My crusade against
New Math (19701982) & Constructivist Math (1988.) in BelgiumFlemish
Introduction: Mathwars: New Math & Constructivist 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 BelgiumFlemish 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
15yearolds', 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 inspectorgeneral 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' newmath' 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 inspectorgeneral of technical education, G. Smets,
wrote to us in a letter in response to the' Mud boatpublication' 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
inspectormathematics 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, VVWLcongress). 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 (19701982)
3.1 Our resistance during the period 19701981
In the years 19681969, 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 KULeuven 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
too 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 threedimensional 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 19781982, 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 and Community,
jg). 28, p. 3536). 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 Mathematics' with a
publication of' New Mathematics: a flag on a mud (muddle)boat' (Education paper
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 Subsequent 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 relationfunction 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' Modderschuit', we extensively illustrated "that
the New Math of the Bourbaki Group could not be separated from the
structuralist and logicalformalistic 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 socalled Bourbaki Group had ambitious plans to describe the full
range of mathematics in a very systematic way, starting from axioms and
the setdoctrine. 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 logicalformalistic approach
In our publication of 1982 we also showed that
"Bourbakimathematics could not be separated from the structuralist and
logicalformalistic 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 crosssection, 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. WienerKreis on. Prof. Apostel
sought for formallogical 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
interwar 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
logicalpositivists 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
logicaldeductive 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 logicaldeductive buildup
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 crossset (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 logicalhierarchical 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: parallelismevenness 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 multifaced 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
socalled "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
DirectorsGeneral and inspectors, were
silenced from above. The
DirectorGeneral 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 selfcensorship are greater than ever. We have also noted
this recently in connection with the M decree.
The New Mathnew
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: constructivist mathematics, which shows
little appreciation for mathematics as a cultural discipline.
