Inventing modern mathematics:

1. Geert Vanpaemel, Dirk De Bock, and Lieven VerschaffelSecond International Conference on the History of Mathematics EducationLisbon, Portugal, October 2-5, 2011 Inventing modern mathematics: Early debates on mathematical curriculum reform in Belgium (1950-1968)

2. Introduction • (Very) new and ongoing research project. • Motivation: Belgian mathematicians and math teachers played an important role in the (international) “modern mathematics” reform movement that had its peak in the sixties and seventies of the previous century. • So far, no systematic research has been undertaken with respect to the origins of this movement in Belgium from an international perspective.

3. Introduction • Key figures of this reform movement: Georges Papy (mainly for secondary education) and Frédérique Lenger (mainly for elementary education and Kindergarten). • But, G. Papy – who was a professor of mathematics at the Université libre de Bruxelles – only got involved around 1958... • Our main research intrest: early debates (starting around 1950) which shaped this drastic educational reform in its early stages in an international context.

4. Servais • A major personality promoting mathematics education reform in Belgium from the beginning of the 1950’s was Willy Servais (1913-1979). Dieudonné, Frédérique Lenger, Willy Servais and Lucienne Félix at the CIEAEM meeting (“rencontre”) in La Rochette (Melun), 1952

5. Servais • Who was Willy Servais? • Mathematics teacher (and later study prefect) at the Athénée du Centre à Morlanwelz (French speaking part of Belgium) • First president (1953-1969) of the Belgian Society of Mathematics Teachers • Key figure in the early years of CIEAEM of which he became, very soon, the secretary and later the president • Member of the first editorial board of ESM

6. Servais • Who was Willy Servais? • Servais also attended and played an important role in the first ICME conference in Lyon (1969) A. Z. Krygowska, H. G. Steiner, F. Lenger, W. Servais at ICME Lyon

7. Servais • Who was Willy Servais? • Servais was a “gentleman”, an open-minded personality who rather tried to reconcile than to divide people (opposite to the latter G. Papy, who is often described as rather “rude” or even “arrogant”). • Fraction between Papy/Lenger and Servais around 1970? Papy/Lenger stopped attending CIEAEM (while Servais didn’t) around 1970, whereas Servais kept good and strong contacts with Hans Freudenthal (even during the 1960’s and 1970’s)

8. Servais • How Servais’ ideas were formed? • We know little about Servais’ formative years, as well as about his professional career in Morlanwelz. • Enlisted in the Belgian army at the beginning of World War II, and deported as a prisoner of war to the German camp in Tibor. • During that period, Servais read several books on mathematics, logic and methodology, in particular by the Swiss psychologists and philosophers Piaget and Gonseth, and he also read the first books of Bourbaki

9. Servais • How Servais’ ideas were formed? • Upon his return in Belgium in 1945 Servais got involved in debates on math education in primary schools • Likely, this involvement is related to the promotion of the colored Cuisinaire rods, invented by Georges Cuisinaire (1891-1976) who worked as a primary teacher in Thuin

10. Servais • How Servais’ ideas were formed? • Through Louis Jéronnez, the director of his school and mathematician too, Servais was invited to the 1951 meeting of CIEAEM • This meeting is pivotal in the development of Servais’ views on mathematics education: he found a wider field of interest in the renewel of mathematics education • Servais came in close contact with other key personalities in math education of that time (Gattegno, Choquet, Dieudonné, ...)

11. Servais • Servais’ main ideas • Generally speaking: • In the early 1950’s: promotion of concrete didactical approaches to mathematics education (e.g. Cuisinaire rods, use of films, truth tables, ...) and plea for the introduction of new contents (probability, statistics, math related to “calculation machines”, ...). • Likely, around the mid 1950’s, Dieudonné introduced/got acceptance of the Bourbaki ideal of a unified mathematcs in CIEAEM (and influenced Servais and – around the end of the 1950’s – strongly influenced G. Papy).

12. Servais • Servais’ main ideas • In more detail: • Strong belief in mathematics as an indispensible tool for modern citizens • “Our time marks the beginning of the mathematical era” (Editorial of the first issue of Mathematica & Paedagogia, 1953) • “It is convenient to be proud about a mathematical tradition, it is also important to permet “the mathematisation” that will come” (same source)

13. Servais • Servais’ main ideas “These students, our students, who are now working in our classrooms, will become, tomorrow, mathematicians, physicists, officers, technicians or, simply, representants of hundreds of professions in which mathematics will play a more efficient role” (Servais, 1954)

14. Servais • Servais’ main ideas • And for understanding (new developments in) other scientific disciplines • “mathematics in the form of statistics and probability theory has become crucial to biology, sociology and economics. It has become an important instrument for logical research” • “If mathematics could be a goal in itself, for an ideal of abstract beauty and logical formation, it’s also the servant of all sciences to which it can offer its formal framework (“ossature”)

15. Servais • Servais’ main ideas • A universal language for understanding and “conquering” our world • With respect to quantum physics: “the mathematical symbolism is the only way of understanding and prevision, which connects human intelligence to the deep realities of the atomic world” (Servais, 1967)

16. Servais • Servais’ main ideas • As a consequence: Strong plea for international collaboration • “La mathématique, langue vraiment universelle a, par sa nature, une vocation internationale ; nous ouvrirons nos colonnes à nos collègues des autres pays” (Servais, 1954)

17. Servais • Servais’ main ideas • Attention for new developments in mathematics • The basic notions of mathematics are number and space, but now “a new mathematics is opening up, the mathematics of abstract spaces, of general topology and modern algebra. Each of these disciplines leaves undefined the mathematical objects which they study, in order to concentrate its efforts on the study of the relations between these objects and the operational properties of its relations” (Servais, 1953)

18. Servais • Servais’ main ideas • Plea for bridging the gap between between school mathematics and mathematics as a vivid scientific discipline • “The children who enter today into primary school will have a delay of ten to twenty years at the end of their studies […] with regard to the world that they have to understand and to conquer. […] We have to form them today for them to be ready tomorrow” (Servais, 1967) • “It was therefore necessary that the teaching of mathematics would bring the student into closer contact with the world of modern mathematics” (Servais, 1967)

19. Servais • Servais’ reform ideas are in line with the epistemological doctrine of Gattegno, the genetic psychology of Piaget and Gonseth, or the unification (of mathematics as a scientific discipline) of members of the Bourbaki group. • Servais himself was essentially interested in the improvement of school mathematics; in his writings, he had less attention for the philosophical, psychological or mathematical underpinning of his ideas.

20. Another important player of the early fifties: Frédérique Lenger (see: http://www.rkennes.be/) • 1921-2005 • University trained mathematician • Research assistent of professor Libois (ULB) • Involved in teaching at a Decroly school and involved in teacher training • Early member of CIEAEM • President of GIRP (1971-1981) • Maried Georges Papy in 1960 (and became “Frédérique Papy-Lenger”) • Made a doctoral dissertation under the supervision of her husband in 1968 (on early math education and on a methodology for inservice training)

21. From left to right: Teresa Dutra (Portugal), Georges Papy, Frédérique Papy-Lenger at a GIRP meeting in Dubrovnik (1977) GIRP: Groupe International de Recherche en Pédagogie de la mathématique, founded in 1971

22. At the end of the fifties, Servais and Lenger collaborated for an experimental program of “modern mathematics” (for a school for Kindergarten teachers!) and asked the advice of Georges Papy (°1920) Papy, heavily leaning on Bourbaki, radically changed the impetus and the orientation of the new math movement in Belgium His approach – abstract, structural, algebraic – imposed a strict format of mathematics teaching

23. Did Servais’ (and other reformers’) plea for reform mean that math education in Belgium was deficient at that time? • Probably not! • Comparative UNESCO study (1959-1961) in 12 European countries with 13-year olds: Belgium stood out with the highest scores for mathematics. The scores were “particularly high for items which require reasoning and the use of concepts” • Since the 1930’s: Belgian primary schools were considered as a model of child-centered pedagogy (as introduced by O.Decroly) • Educational quality was systematically studied and monitored by school psychologists (who developed different types of tests)

24. Hotyat • Were Servais’ ideas in line with these of Belgian school psychologists and educators of that time? • No! • From international studies, Hotyat (1961) concluded that generalized and symbolic reasoning was usually not attained by pupils before the age of 13. Younger pupils would still return to a more practical level to solve analogous problems with a trial and error approach • Hotyat showed himself critical about what he called the “audacious experiments” with the introduction of modern mathematics in primary school curriculum. These experiments were in his opinion inspired by the views of (academic) mathematicians, which he distinguished from the school teachers of mathematics.

25. Hotyat • To him, the efforts of teachers regarding innovating didactical approaches lacked the control by objective measurement, necessary to build a truly experimental science. Their personal experiences led to a form of ‘experiential’ (or experience-based) psychology, but did not contribute to ‘experimental psychology’. • But he acknowledged that psychological theories were not in accord with each other, and had still very little to offer as a guiding principle for actual didactical reform. • And he recognized that the reform initiatives, taken by individual schools or teachers, were symptoms of a “profound movement”, and he conceded that the first results were indeed encouraging.

26. To better understand Hotyat’s critique (and also some of Servais’ quotes): A note on the three types of teachers and their education in Belgium at that time • Primary school teachers: lower secondary education (until the age of 15) folowed by a special training program of 4 years (that did not give entrance to university!). • Lower secondary school teachers: secondary education (until the age of 18) folowed by a non-university program of 2 years. • Higher secondary school teachers: secondary education followed by a full university training and (a limited) additional pedagogical program. • → All main protagonists of the reform belonged to the third category of teachers

27. Servais • As a consequence: • Servais’ (and other reformers’) view on math education was defined by their schooling first as a mathematician. • Their teaching experiences were confined to the higher level (16-18 year). • They were (only) in contact with the “elite” preparing them to higher education (the age of obligatory schooling was set at 14 at that time)

28. Likely, the reform debate in mathematics education was strongly related to two other debates: • The emancipation of non-university teachers and the professionalisation of their status (most of the proposed changes in mathematics concerned the first three years of secondary education) • The emancipation of mathematics (in that time, Latin-Greek was the most prestigious track in Catholic secondary schools in Belgium)

29. The reform movement was strongly supported by the Université libre de Bruxelles (Libois, Servais, Lenger, Papy), and thus reflected a “liberal” philosophic orientation, highly estimating science and mathematics (in contrast with more humanistic ideals and orientation towards of the catholics). The political atmosphere in Belgium at that time, with a strong tension between catholic and state school, was turbulent (leading to a “school pact” in 1958)

30. Some tentative conclusions • All main protagonists in the reform movement of mathematics education in the 1950’s and 1960’s were university trained mathematics teachers • Before Papy entered (and dominated the debate), reform ideas were rather “moderate” and the reformers did not explicitly strive for the Bourbaki ideal of a unified mathematics • Mathematics reform debate was related to other educational debates of that time