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My crusade against
New Math : 1972-1998 in Belgium-Flemish
schools
Introduction:
Math-wars: 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
Belgium-Flemish in 1982
Exactly 41 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 barge, Onderwijskrant nr.
24 www.onderwijskrant.be). 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 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 Louvain professors Roger Holvoet
and Alfred Warrinnier, from inspectors who had participated in new mathematics
methods ...
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 our publication' 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 sends 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 opposed the introduction
of NM in education. Dutch prof. Hans Freudenthal e.a. 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 Piet 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 1970-1981
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 teh formalistic & abstract Naw Math.
In the early seventies we did our best to convince those responsible for the
educational umbrella organisations to not introduce the 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 & Gemeenschap. 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 New Mathematics, I was one of the autors. 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 New Math:
too
formalistic, heavenly (floating) mathematics * too early abstraction * too
much verbal bullying and verbal ballast to the detriment of the application
aspect of mathematics (calculating, memorizing, automatistion...)
*little respect for the classical discipline of mathematics
as a cultural product. *Not achievable for many students: too many
pupils have to switch to special needs education after the third year of primary school. *Many parents were 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 for the primary school.
3.2 Mathematics
campaign 1982: NM: a flag on a mud barge 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 De Groote further
fantasized that the less gifted students now have a better place for
themselves (Person & Gemeenschap 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 NM with a publication
of' New Mathematics: a flag on a mud
barge/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
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 professors mathematics: Conversion Louvain prof. 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 Louvain 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 publicaton of1982 , 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 Louvain
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 NM'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.
Traditional notions 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 sry 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 Piet Vermeylen and topofficials) 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. We have also noted this recently in connection with the M
decree.
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.
Note: In the first degree secondary school
unfortunately, they opted in 1997 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.
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