Dirk De Bock
email: dirk.debock@kuleuven.be

The reception of modern mathematics in Belgian technical education: Adhering or resisting?

Keywords: Belgium, MATEC, modern mathematics, reform movement, technical education

Belgium was one of the first European countries to be involved in the modern mathematics movement of the 1960s. The reform was led by a university professor (Georges Papy), supported by a number of enthusiastic teachers of general secondary education. The reformers showed almost no interest in technical education. In the mid-1960s, however, it became clear that also the mathematics programs of technical secondary education would be modernized. We examine the diverse reactions of stakeholders of technical education to the announced reform. Emile Ridiaux and like-minded teachers prepared and attempted to use modern mathematics for technical applications. Others, e.g.

Introduction

Belgium was a pioneering country in the modern mathematics movement of the 1960s. A first experiment with modern mathematics began as early as 1958, one year before the landmark Royaumont Seminar. The experiment was carried out with prospective kindergarten teachers and was led by Frédérique Lenger and Madeleine Lepropre (De [START_REF] Bock | Early experiments with modern mathematics in Belgium. Advanced mathematics taught from childhood?[END_REF]. For technical assistance with follow-up experiments, the experimenters enlisted the help of Georges Papy, a professor of algebra at the University of Brussels. However, Papy did not limit himself to consulting: From 1959 he started his own experiments and would completely dominate the movement. In May 1961, he founded the Belgian Centre for Mathematics Pedagogy that grouped the country's most prominent modern mathematics enthusiasts. The Centre coordinated all aspects of the reform movement: It edited new curricula, set up experiments, developed course materials, retrained teachers, established local working groups, etc. The interpretation of modern mathematics was quite radical. Sets, relations, and algebraic structures formed the basis; logic, deductive reasoning, and proof were central. With appropriate political support, Papy's modernization efforts would lead to a mandatory implementation of modern mathematics in the general sections of secondary schools from September 1968.

Papy and his team members, who typically had an academic or general education background, showed little interest in technical education. No experimental programs or experiments were anticipated for this type of education. From official directives of the mid-1960s, however, it became clear that technical schools would also be involved in the modernization of the mathematics programs, a reform that prioritized intellectual formation over practical utility. A circular letter issued by the Nationaal Verbond van het Katholiek Technisch Onderwijs [National Association for Catholic Technical Education] on November 15, 1966, stated that every mathematics teacher in technical education should urgently inform himself about modern mathematics. To this end, these teachers were strongly advised to attend the classes given at the initiative of the Belgian Centre for Mathematics Pedagogy, and school boards were asked to give their teachers the opportunity to do so. [START_REF] Holvoet | De werkgroepen van het Centrum[END_REF]. We find a similar recommendation in the 1966 mathematics program of the Enseignement technique et professionnel féminin de l'État [Technical and professional education for women in the state], as cited in [START_REF] Holvoet | De werkgroepen van het Centrum[END_REF]: "All teachers are strongly advised to increase their knowledge by all means and in particular to attend regularly the courses organized by the Belgian Centre for Mathematics Pedagogy1 " (p. 101).

In this paper, we search for answers to the following research question: "How have teachers and other stakeholders in technical education responded to an announced reform that seemed diametrically opposed to the goals of mathematics in technical education?" To answer this question, we have analyzed the available reform documents and all Belgian journals for mathematics teachers from that period: Mathematica & Paedagogia, Nico, and Matmo (see below). Moreover, we systematically searched BelgicaPress, a digital database of historical Belgian newspapers up to 19702 . This study is part of a line of research on the history of the modern mathematics movement in Belgium that recently led to the book Rods, sets and arrows: The rise and fall of modern mathematics in Belgium (De Bock & Vanpaemel, 2019). However, the reform in technical schools and the role of key actors in it remained underexposed. The present study aims to fill this gap in the existing literature on the Belgian modern mathematics movement. The scope of this type of research is, however, not just "purely" historical: Researching mathematics education of the past can provide historical insights that support the future, or as [START_REF] Van Bendegem | Review of the book Rods, sets and arrows: The rise and fall of modern mathematics in Belgium, by D. De Bock & G. Vanpaemel[END_REF]

Ridiaux's actions to welcome modern mathematics

Emile Ridiaux (1924Ridiaux ( -2006)), trained as an engineer, was a mathematics teacher in the secondary technical section of the Université du Travail [University of Labor] in Charleroi, an important industrial center in the French-speaking part of Belgium. With a view to preparing mathematics teachers in technical schools for the arrival of modern mathematics in their classrooms, Ridiaux founded a new teachers' journal, titled Matmo, a reference to "modern mathematics", and subtitled Revue des ensemblistes du secondaire [Journal for enthusiasts of sets in secondary schools]. Matmo was published between 1964 and 1968. A total of thirty issues appeared: Ten in the first volume (1964-1965), ten in 1966, five in 1967, and five in 1968. Layout and style of the journal were very basic and each issue had only 16 stapled pages. The journal did not question the upcoming reform as such, on the contrary. In the first Editorial (September 1964), we read:

We do not claim to be scholars! We know the scholars, because they have come down to us, to our teaching! At the secondary level, even at the primary level, we want to apply their advice, because we have confidence, and, with them, we have hope in a much better mathematical future.

Browsing through the different issues of Matmo, one notices that most of the contributions were written by Ridiaux himself and some of his colleagues. Mainly lesson ideas for lower (technical) secondary education are presented, on "modern" topics such as sets, relations, and structures (inspired by Papy's proposals), but also "classical" topics from algebra, higher arithmetic, and geometry are considered from a modern point of view. Sometimes student reactions are included in the lesson plans, showing that the lessons had already been trialed. Attention is also paid to applications of mathematics, especially from physics (e.g., electrical circuits), but also from chemistry and biology. Matmo regularly informs its readers of new developments related to mathematics education, both in Belgium and abroad (e.g., with echoes from France and the USA). Specific Belgian reform initiatives, in particular the in-service courses of Papy's Centre, receive ample attention. The emergence of an opposition movement in 1966 (see the next section) is also reported and limitedly documented, but already in the first issue of 1967, Ridiaux writes that although he welcomes remarks and criticisms, "We would like to avoid a quarrel between the ancients and the moderns, especially among teachers..." (in capitals, no pagination). After that, Matmo will not include any news about the opposition movement… It is worth mentioning that in 1969 Ridiaux published a textbook on a remarkable method in which arrow graphs, one of the favorite representational tools of Papy and his collaborators, were used to transform algebraic formulas [START_REF] Ridiaux | Transformations de formules par la méthode des graphes. Wesmael-Charlier. Ridiaux, E[END_REF]. This technical skill was not considered important by modern mathematics reformers, but it was fundamental to (technical) applications of mathematics. In Ridiaux's method, a path of arrows, representing operations, is drawn from a variable to be found to a given variable. By tracing the inverse arrows, which represent the inverse operations, the requested variable can be calculated (see Figure 1). Sometimes, Ridiaux also visualized the different steps in this process via film strips, a didactic tool known from Papy's Mathématique Moderne (De Bock & Vanpaemel, 2019). In [START_REF] Ridiaux | Transformations de formules par la méthode des graphes. Wesmael-Charlier. Ridiaux, E[END_REF], the author explained how this method can also be used to solve (systems of) linear equations. Ridiaux's method is no longer used in Belgian schools today, but it is still recommended, albeit in a modified form, by some mathematics educators of the 21st century (see, e.g., [START_REF] Noël | Pour une approche TGF dans les logiciels didactiques[END_REF]. The case illustrates how Ridiaux tried to didactically modernize mathematics for technical education, his way of conforming to reformers' expectations. in which he argued for a reform but one different from that proposed by Papy and his Centre [START_REF] Schwilden | L'enseignement technique « oui » à la réforme[END_REF]). More specifically, Becquet advocated an approach that "makes better use of intuition, repetition, and new methods, such as those used in programmed courses". Becquet did acknowledge, however, that it would be desirable "to integrate the useful parts of new theories, taking into account the various levels and objectives of education" (Schwilden, 1968, p. 7). He also insisted that the reform be carried out slowly and gradually, and that the old not be swept away at once:

Basically, the first phase would be focused on setting the traditional in order and the second phase on implementing the new theories. In this way, there would never be a deep conflict in the continuity of the curriculum, but a slow and secure evolution. The teachers themselves would not feel like pawns, like toys in the hands of curriculum developers. (Becquet in Schwilden, 1968, p. 7) As early as September 1963, Becquet and some like-minded mathematics teachers at schools for technical education launched their own experiment to modernize the mathematics curricula for 12to 18-year-old students in these types of school [START_REF] Gadeyne | Sur le front des maths[END_REF]. Eventually, 35 schools would follow Becquet's experimental method [START_REF] Vermeulen | La mathématique, cette inconnue… Le Soir[END_REF]. Becquet considered his experiment successful: A survey of participating teachers revealed that the method better established basic knowledge and facilitated understanding of mathematics. As a result, fewer students failed mathematics than before (Le Soir, April 24, 1968) The technicians insist on the practice of calculation. A solid knowledge of the fundamental operations involving integers, decimal numbers, and fractions is essential. The study of the metric system and its elementary applications, initiated in elementary school, must be continued. The drawing activity must therefore, in principle, be based on a prior knowledge of geometric notions acquired by experimental or intuitive means in the course of mathematics. It should be noted that the first-year geometry course should, more than in general education, introduce students to the knowledge of the common plane forms and solids and train them to see in space, because these notions and this skill are indispensable for the drawing of parts in the wood and iron workshop. It follows that the preservation of an elementary practical teaching of geometry, but always with a formative character, is essential. (Matmo 10/66, no pagination)

As an alternative, MATEC proposed that the lower grades of (technical) secondary education would study arithmetic, both calculation techniques and problem solving, the metric system, the basics of algebra (first degree algebra and the roots of a quadratic equation), plane geometry, trigonometry with right triangles (perhaps extended to non-right-angled triangles), and the first elements of solid geometry. Becquet did not oppose, at some later stage, the integration of some elements of vector and matrix algebra in schools for technical education. As stated by [START_REF] Noël | La réforme des maths moderne en Belgique[END_REF] and evident from the above listing, MATEC considered mathematics primarily as a technical tool and not as gymnastics of the mind. In the textbooks for Becquet's experiments, developed under his supervision by a group of teachers from technical schools involved in these experiments, the difficult chapters were split into two parts to separate the meaning of operations from the practice of calculation. Needless to say, the work of MATEC was strongly criticized and even ridiculed by Papy ("Mr. Becquet, the courses of Belgian Centre for Mathematics Pedagogy are open to you.

Learn! You will be forgiven a lot", Papy, 1968, p. 34).

Vain hopes and math wars in the late 1960s...

The political decision to mandatorily introduce modern mathematics in the first years of secondary education from September 1, 1968, was made by Henri Janne, Minister of Education in an outgoing government, and announced in a circular of May 14, 1965 [START_REF] Janne | Programme optionnel de mathématique (ministerial circular)[END_REF]. Janne was a former rector of the University of Brussels, a socialist and political friend of Papy. On April 11, 1968, Janne's decision was confirmed by Frans Grootjans and Michel Toussaint, the Ministers of Education at the time, but their decision only concerned the general sections of secondary education [START_REF] Grootjans | Leerplan wiskunde voor het eerste leerjaar van het rijksmiddelbaar onderwijs van de lagere graad/Programme de mathématique pour la première année d'études de l'enseignement moyen du degré inférieur[END_REF]). Thus, no formal decision was yet made about modern mathematics in technical secondary education. Moreover, on June 17, 1968, a new government was formed with two new Ministers of Education: Piet Vermeylen and Abel Dubois. The newly appointed Ministers soon allowed some schools of general secondary education "not yet ready to implement the new curricula" to postpone the introduction of modern mathematics by one year [START_REF] Noël | La réforme des maths moderne en Belgique[END_REF]. Could the decision of Grootjans and Toussaint be reversed? A time of uncertainty followed for Papy's partisans, a time of hope for his opponents. This uncertainty led to a fierce math war in 1968-1969, especially in French-speaking Belgium (De Bock & Vanpaemel, 2019).

During that war, MATEC, which in the meantime had grown to a grouping of more than 400 school principals and teachers (Le Soir, April 24, 1968) [START_REF] Derwidué | Ouverture de l'assemblée de contestation[END_REF]. In reaction to this opposition, a group of teachers who supported Papy's reform project for the secondary level was created-the Comité pour la Promotion de l'Enseignement Mathématique [Committee for the Reform of Mathematical Education]) [START_REF] Genaert | La guerre des « maths ». La préfète Andries, de Bruxelles, fonde un comité de promotion de la mathématique moderne[END_REF]. In 1968-1969, both organizations mobilized all those involved in the reform, including parents, for large-scale information meetings, hearings, and colloquia held in major Belgian cities (Figure 2). The war was eventually won by Papy and his proponents. Minister Dubois, also a socialist, stuck to the decision of his predecessors. In an address delivered at the teaching college in Nivelles on April 27, 1969, he stated:

The new curriculum is the Belgian version, very pragmatic and very adaptable, of a mathematical conception which is now being introduced in all industrialized countries; ... it constitutes a clear obligation for all schools run by the state; no one, my predecessors nor myself, has ever envisaged reconsidering it. (Dubois, 1969, p. 3) When from September 1, 1969 all students of the first years of secondary education, both in the general and in the technical sections, without any exception, were subjected to modern mathematics, the late 1960s math war ended quickly. The official programs of the technical sections were similar to those of the general sections except for a few details. For Dubois, this was a deliberate decision, even a matter of principle, viz. an occasion to eliminate, maximally, divisions between different types of education (and thus to upgrade the status of the technical schools).

It would be unacceptable if not [the same programs were applicable] and that, with the introduction of the new program, the harmful division between general and technical education, each in their respective fortresses, would be continued. (Dubois, 1969, p. 3)

Conclusion

In the Belgian modern mathematics movement of the 1960s, technical secondary education entered the picture relatively late. The technical schools did not ask for the reform, but when it became clear that they would be involved, reactions were mixed. Ridiaux and some like-minded teachers adhered and tried to make the best of it, explaining in a newly founded journal Matmo how to incorporate modern elements into mathematics for technical education; others, such as Inspector Becquet and the members of MATEC, chose to resist. After a bitter math war in 1968-1969 between supporters and opponents of Papy, especially in French-speaking Belgium, modern mathematics was also implemented in technical secondary education, one year later than in general secondary education but with an almost identical program.

Future research could clarify the specificity of the Belgian modern mathematics movement in an international context. However, despite increasing scholarly interest and some single analyses with a specific scope (e.g., [START_REF] Kilpatrick | The new math as an international phenomenon[END_REF]Vanpaemel & De Bock, 2019), comparative research on the international "New Math" phenomenon is still in its infancy.

Figure 1 :

 1 Figure 1: Example of Ridiaux's method for the transformation of formulas using arrow graphs; the variable B has to be calculated from the given formula (Ridiaux, 1969, p. 18) Becquet and the action of MATEC From Charles-François Becquet (1915-1987), a totally different sound was heard. Becquet was a master of mathematics (University of Liège, 1940), a teacher and later an inspector of technical secondary education organized by the state. In addition, though less relevant here, Becquet was a political author and militant in the Walloon Movement during and after World War II. In July 1963, Becquet had published an article entitled Réformons les mathématiques [Let's reform mathematics],in which he argued for a reform but one different from that proposed by Papy and his Centre[START_REF] Schwilden | L'enseignement technique « oui » à la réforme[END_REF]. More specifically, Becquet advocated an approach that "makes better use of intuition, repetition, and new methods, such as those used in programmed courses". Becquet did acknowledge, however, that it would be desirable "to integrate the useful parts of new theories, taking into account the various levels and objectives of education"(Schwilden, 1968, p. 7). He also insisted that the reform be carried out slowly and gradually, and that the old not be swept away at once:

  [...] Vocational training, the practice of technical drawing, and the technology course require real aptitudes in the above-mentioned subjects.[…] In technical schools, the first-year drawing program includes geometric drawing and, already, an introduction to industrial drawing through the representation of simple parts. Teaching recipes alone would have no educational value.[…]

Figure 2 :

 2 Figure 2: Invitation flyer for a gathering of parents in Liège on June 23, 1969The headlines of articles in the Francophone press of the period were telling: "La guerre des maths aura-t-elle lieu?" [Will the maths war take place?] (Spécial, March 6, 1968, p. 16), "Sur le front des maths" [On the maths front] (Pourquoi Pas?, August 29, 1968, p. 105), "Des cobayes pour les Papystes" [Laboratory animals for Papy'ists] (Spécial, April 9, 1969, p. 15), "À quand un cessez-lefeu et une commission d'armistice? Le pénible spectacle offert par la 'guerre des math' " [When will there be a ceasefire and an armistice commission? The painful spectacle offered by the "math war."] (Le Soir, April 27-28, 1969, p. 7), "Nouvelles maths: pour ou contre?" [New maths: for or against?] (La Libre Belgique, June 12, 1969, p. 5). The leftist press was mostly sympathetic to the reform initiated by Papy and his Centre, not only because Papy was a socialist, but also because there was a vague belief in the emancipatory power of the project.

  . In support of his actions, Becquet had founded a new teachers' association, Mathématique et Technique (MATEC) [Mathematics and Technique], in the first half of 1966. Matmo 04/66 recorded a short press release about the initiative:The Board of Directors of the teachers' association Mathematics and Technique expresses its strongest reservations regarding the consequences for the future of students in technical education resulting from the introduction of systematic teaching of set theory in the lower grades of technical and vocational secondary education.

	defense of technical education and its illustration in the framework of mathematics. (no
	pagination)
	A Board of Directors was elected consisting of Becquet (president), J. Loomans (general secretary),
	and G. Benoit (administrative secretary), the latter two being principals of a technical school of,
	respectively, the official and Catholic network. Nine working groups were also established, each to
	address a specific problem related to the mathematics programs, and the relationships between
	technical education, on the one hand, and other forms of education and industry, on the other
	(Matmo 04/66, 05/66). At a session on October 12, 1966, MATEC's Board of Directors adopted the
	following resolution:

About 30 school principals and teachers belonging to the various networks of technical education met, under the presidency of Mr. Inspector Becquet, at the Institut d'Enseignement Technique de l'État, in Namur, to constitute an association Mathematics and Technique, whose aim is the (Matmo 10/66, no pagination) MATEC thus explicitly opposed the systematic introduction into the lower grades of technical secondary education of set theory, "a gateway to philosophy and logic but not to technical calculation", as Becquet had declared in an interview in Le Soir, a leading newspaper in the Frenchspeaking part of Belgium

(Le Soir, April 24, 1968, p. 6)

. But what did the technical schools fear would be lost, and what specifically did they propose as an alternative? The main concern, perhaps, was that in Papy's curriculum proposals mathematics would not be helpful to and would even isolate itself from other courses in technical education. In a letter dated November 1966 from Becquet addressed to the principals of the technical schools we read:

  , further developed into a more or less structured opposition movement against Papy and the Belgian Centre for Mathematics Pedagogy. Already in 1966, it united forces with the Association des Docteurs et Licenciés en Sciences Mathématiques sortis de l'Université de Liège [Association of Doctors and PhDs in Mathematical Sciences from the University of Liège] (Matmo 10/66). Toward the end of the 1960s, an umbrella organization of mathematics teachers opposed to Papy's reform was established-the Association des Professeurs de Mathématique de l'Enseignement Secondaire [Association of Teachers of Mathematics of Secondary Education]

All translations were made by the author.

Collections of the Royal Library of Belgium (https://www.belgicapress.be/). Keywords used: "Becquet", "Matec", "Matmo", "Papy", "Ridiaux".


Wendy Goemans
email: wendy.goemans@kuleuven.be

Dirk De Bock

The "best" axiom system for teaching geometry to secondary school students: A source of controversy in the early 1960s

Keywords: Axiom system, geometry education, Gustave Choquet, Jean Dieudonné, modern mathematics

Jean Dieudonné's "Euclid must go!" provoked controversy among participants at the 1959 Royaumont Seminar. Although Dieudonné's outcry led to misunderstandings, many contemporaries shared his view that geometry education needed to be modernized. On how to proceed, there was less agreement. At international meetings after Royaumont, no clear and supported solution could be proposed. For Dieudonné and other (Bourbaki-oriented) mathematicians, linear algebra was the royal road to geometry, but even then opinions differed widely. The controversy peaked in 1964 with the publication of Gustave Choquet's L'Enseignement de la Géométrie and, shortly thereafter, Dieudonné's Algèbre Linéaire et Géométrie Élémentaire. We carefully analyze these two seminal books and the role they played in the debates on modern mathematics in the early 1960s. Key differences are revealed and put into perspective with the similarities identified.

Introduction

The 1959 Royaumont Seminar is widely recognized as a turning point in the history of the modern mathematics movement of the 1960s [START_REF] Bjarnadóttir | Fundamental reasons for mathematics education in Iceland[END_REF][START_REF] Skovsmose | Towards a critical professionalism in university science and mathematics education[END_REF]. The Seminar brought together American and Western European reformers for the first time, and it also marked the start of this movement in several countries not previously involved. Although in recent years several researchers have argued that very diverse positions were taken at Royaumont, for example about the role of applications [START_REF] De Bock | From Royaumont to Lyon: Applications and modelling during the sixties[END_REF] and of systematic psychological research (Schubring, 2014a[START_REF] Schubring | The road not taken-The failure of experimental pedagogy at the Royaumont Seminar 1959[END_REF], in the collective memory of mathematics educators, Jean Dieudonné's sharp attack on the traditional teaching of (Euclidean) geometry, for which his battle cry "Euclid must go!" was symbolic, continued to resonate.

A lot of ink has also been spilled about the exact meaning of Dieudonné's slogan. Whence this vehement attack on "Euclid" while he deeply admired the achievements of the Greeks to mathematics ("I consider their creation of geometry perhaps the most extraordinary intellectual accomplishment ever realized by mankind", Dieudonné, 1961, p. 35)? First, Dieudonné criticized what he considered a fossilized (French) educational tradition in geometry education, or as Hans Freudenthal (1967, p. 745) formulated it:

The inventor of the slogan "A bas Euclide" a follower of Euclid? It looks odd, but it does so simply because few slogans have been misunderstood as badly as this one. Partly, it was Dieudonné's own fault. When he cried "A bas Euclide," he actually meant "A bas 'Euclide'," (viz. the Euclid of French lycée textbooks), but in oral discussions it is a hard thing to pronounce quotation marks.

Almost everything to which much attention was paid in this tradition, in particular "the triangle", had, according to [START_REF] Dieudonné | New thinking in school mathematics[END_REF], "as much relevance to what mathematicians (pure and applied) are doing today as magic square or chess problems!" (p. 36). However, Dieudonné was clearly in favor of an axiomatic approach to geometry, but not the old one of Euclid with its fuzzy distinction between "axioms" and "postulates", but of a "modern" one based on linear algebra. Dieudonné's views caused "sharp controversy" among the Royaumont participants, yet all agreed that a modernization of geometry education was needed. There was, however, less unanimity on how that modernization should be achieved. In a number of follow-up meetings in the early 1960s, the international mathematics education community attempted to come to some form of agreement, though without success. The debate quickly rigidified into the search for the "best" axiom system for secondary school geometry, and further hardened with the publication of Gustave Choquet's L'Enseignement de la Géométrie and Jean Dieudonné's Algèbre Linéaire et Géométrie Élémentaire, both in 1964. In this paper, we first provide a brief overview of the development of the geometry debate in the first half of the 1960s. Second, we carefully analyze the aforementioned books by Choquet and Dieudonné in order to unravel the differences and commonalities in vision of these two scholars. To this end, inspired by the topic model for discourse analysis of [START_REF] Jacobs | Topic models meet discourse analysis: A quantitative tool for a qualitative approach[END_REF], we first manually compiled a list of topics (such as for instance "droite" [straight line], "symétrie" [reflection], …) that appear in the books, and this page per page. Then, we counted (using the freely available online tool of browserling on https://www.browserling.com/tools/word-frequency), how many times a topic is mentioned in each book. We compared how the topics that ranked highest in this count were dealt with in the two books. In comparing the manipulation of the different topics, we paid specific attention to the differences caused by the axiom system chosen. In the final section of this paper, we discuss how it came to a settlement, at least between Choquet and Dieudonné.

The geometry debate in the aftermath of Royaumont

The first half of the 1960s was particularly rich in international activities on the modernization of school mathematics. In addition to the annual meetings of the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM), important international meetings were held in, among other places, Aarhus, Denmark (1960), Zagreb-Dubrovnik, Yugoslavia (1960), Bologna, Italy (1961), Budapest, Hungary (1962), Athens, Greece (1963), and Frascati, Italy (1964). These meetings were initiated by one of the following international bodies, either in partnership or not: The International Commission on Mathematical Instruction (ICMI), the Organization for European Economic Cooperation (OEEC; in 1961 joined by nations outside Europe to form the Organization for Economic Cooperation and Development, OECD), and UNESCO. A detailed discussion of the objectives and conclusions of these meetings is beyond the scope of this paper and for that we can refer the reader to, for example, Furinghetti and Menghini (to appear). To indicate the international context and "climate" in which the polemic between Choquet and Dieudonné occurred, we discuss in a nutshell some key elements related to the debate on geometry education.

Although the 1960 Aarhus meeting was intended to have a broader scope, the debates concentrated on "Modern teaching of geometry in secondary schools with particular emphasis on ways of treatment opened up by developments lately, in particular by the algebraization of mathematics" (Behnke et al., 1960, p. ii). No less than five different proposals for "new" axiom systems for secondary school geometry were presented, including by the Royaumont lecturers Dieudonné and Choquet. Dieudonné reiterated his position that geometry education should be approached from the study of two-(or three-) dimensional vector spaces over the field of real numbers, equipped with an inner product. Choquet proposed his axiom system for affine and metric plane geometry that he would adopt in L'Enseignement de la Géométrie (1964). Although views differed, there seemed to be a consensus that Euclidean geometry was too difficult for 12-13-year-olds, even in the form of Bewegungsgeometrie [motion geometry], a German approach rooted in the work of Felix Klein and proposed at Royaumont by Otto Botsch. There were timid attempts, by Freudenthal, Willy Servais, and Tullio Viola, to include psychological aspects in the debate on school geometry, but this led to no less than a quarrel between Freudenthal and Dieudonné, with the latter stating, "la psychologie, je m'en fiche" [psychology, I don't care] (Behnke et al., 1960, p. 104).

In Zagreb-Dubrovnik (1960), a group of experts was commissioned to work out a detailed synopsis for modern secondary school mathematics "in the spirit of Royaumont". The experts agreed on the introduction to set theory, algebra, analysis, probability theory and statistics, but regarding geometry education, the outcome was an ambiguous compromise. For the final years (15-18-yearolds), an axiomatic and structural approach-heavily influenced by Choquet's vision-was recommended, while in the early years (11-15-year-olds), the emphasis would be put on a more intuitive approach [START_REF] Oeec | Synopses for modern secondary school mathematics[END_REF]. The attention to intuitive geometry was largely due to the intervention of the Belgian Paul Libois, a lifelong advocate of intuition-based geometry teaching, but a dissident voice in the 1960s debates (Vanpaemel & De Bock, 2017).

The 1961 meeting in Bologna was intended to discuss the results of Aarhus and Zagreb-Dubrovnik, which had not led to any form of agreement. The discussion in Bologna led to some kind of compromise around a proposal by Emil Artin to define the Cartesian plane axiomatically as a vector space of dimension two with an inner product, but Freudenthal, Libois, and Viola partly disagreed. According to the first, the geometry programs proposed in Aarhus and Zagreb-Dubrovnik were based on an "anti-didactic inversion", expressing that an end product of mathematical activity, the most recently composed structure of mathematics, is taken as a starting point for mathematics teaching [START_REF] Freudenthal | Enseignement des mathématiques modernes ou Enseignement moderne des mathématiques[END_REF].

At the meetings in Budapest (1962), Athens (1963), andFrascati (1964), results of early experiments with modern mathematics could already be presented. In particular, the audacious approach of the Belgian Georges Papy, combining mathematical rigor with an innovative pedagogy, received considerable attention and appreciation.

Two notable books shaping the debate of the mid-1960s

When reading Choquet's L'Enseignement de la Géométrie and Dieudonné's Algèbre Linéaire et Géométrie Élémentaire, one immediately notices the points of agreement in the objective Choquet and Dieudonné aim to achieve with their work; similarities in their views on the teaching of geometry are already apparent in the introduction to their books. Indeed, both authors state that they mainly target secondary school teachers. Moreover, they agree that the teaching of geometry for young children (age less than 13) should be based on observation and experiments, while in the last two or three years of the secondary school, linear algebra should be the basis of geometry. Between these two age groups, children should be gradually familiarized with deductive reasoning.

As (pure) mathematicians, Dieudonné and Choquet were "Bourbakists". Bourbaki's approach to mathematics was axiomatic-deductive, formal, and uncompromisingly rigorous, deliberately excluding diagrams and external motivations. Both our protagonists were convinced that the Bourbaki model for structuring mathematics as a science was also the "best" model for structuring mathematics education, in casu geometry education, with the axioms of linear algebra as a fundamental model. Before presenting their "best" axiom system for teaching geometry to secondary school students, Choquet and Dieudonné digress on how it should not be done, both referring to colleagues along the way, anonymously or not. Choquet seeks to minimize the number of axioms: mais je ne pense pas qu'il soit désirable, comme l'ont préconisé certains professeurs, de prendre au départ de très nombreux axiomes [but I do not think that it is desirable, as some professors have advocated, to take very many axioms at the start1 ] (Choquet, 1964, p. 10) Dieudonné denies the claim that the axioms of linear algebra are too abstract, and refers to Choquet on the way: partir d'un autre système d'axiomes, […] le plus connu est sans doute le système d'axiomes proposé récemment par Choquet, d'une remarquable ingéniosité qui témoigne du grand talent de son auteur, mais que je tiens pour parfaitement inutile et même nuisible. [starting from another system of axioms, […] the best known is undoubtedly the system of axioms recently proposed by Choquet, of a remarkable ingenuity which testifies to the great talent of its author, but which I consider to be perfectly useless and even harmful.] (Dieudonné, 1964, p. 17) However, both authors agree on the fact that the "best" axiom system should separate affine and metric notions, or as Dieudonné phrases it: Je pense surtout ici à la distinction (clairement sentie depuis Poncelet) entre propriétés géométriques de nature « affine » et propriétés de nature « métrique ». Il est particulièrement choquant, du point de vue logique, de voir mélanger en une incroyable salade ces deux types de propriétés dès le début de la traditionnelle « Géométrie euclidienne », mettant exactement sur le même plan des notions aussi différentes que celle de parallèle et celle de perpendiculaire [I am thinking here above all of the distinction (clearly felt since Poncelet) between geometric properties of an "affine" nature and properties of a "metric" nature. It is particularly shocking, from a logical point of view, to see these two types of properties mixed together in an incredible salad from the very beginning of the traditional "Euclidean geometry", putting exactly on the same level notions as different as that of being parallel and that of being perpendicular] (Dieudonné, 1964, p. 13) Likewise, both Choquet and Dieudonné first treat the two-dimensional geometry of the plane before generalizing the concepts to the three-dimensional geometry of the space.

Faithful to his opinion that geometry should start off with a limited number of strong axioms, Choquet immediately sets off with axioms on a straight line and on a set of straight lines. He defines the plane as a set, equipped with a structure, induced by a set of subsets, called straight lines. Then he defines parallel straight lines and he states axioms on the existence of a unique straight line connecting two different points in the plane, and on a unique straight line parallel to a given straight line and passing through a given point of the plane (the so-called parallel postulate). Being parallel is an equivalence relation on the plane, leading to the concept of the direction of a straight line as the equivalence class to which the straight line belongs.

Dieudonné, however, first dedicates some pages to the necessary basic properties of real numbers and to the axioms of Euclidean geometry. He defines the plane as a set on which there exist three operations: addition of two elements, scalar multiplication of an element with a real number, and scalar product of two elements. Assuming the former two operations satisfy the necessary conditions, he calls this plane as soon as possible a vector space. Dieudonné then defines a straight line as an affine subspace with its direction fixed by the corresponding linear subspace (i.e., a straight line through the origin). Two straight lines are parallel if they have the same direction (that is, if they are both affine subspaces resulting from the same linear subspace). From these definitions, and using the properties of the underlying vector space, the proposition that there exists a unique straight line through two different points, and the parallel postulate, are proved.

In conclusion, because of the different choices for the axiom systems in the two books, properties that are put forth as axioms by Choquet can be proved as corollaries by Dieudonné, and vice versa.

In the further treatment of affine geometry in the books by Choquet and Dieudonné, their different starting points lead to only minor or mere formal differences. Noteworthy however, is the topic of the addition of two vectors. In Dieudonné's exposition, this addition is inherent to an operation on the elements of a vector space. Choquet, however, defines this operation by making use of parallelograms: The addition of two points x and y is the point z such that the origin, x, y, and z form a parallelogram. It illustrates Choquet's statement from his introduction, expressing how vectors are naturally at the foundation of geometry: L'axiomatique d'Euclide-Hilbert est basée sur les notions de longueur, d'angle, de triangle. Elle cache à merveille la structure vectorielle de l'espace […]. Le fait qu'un triangle soit la moitié d'un parallélogramme n'a pas empêché qu'on mette l'accent pendant plus de vingt siècles sur l'étude détaillée […] des triangles […]. On voyait le triangle, mais non le parallélogramme qui aurait pu conduire aux vecteurs. [The axiomatic system of Euclid-Hilbert is based on the notions of length, angle, triangle. It hides wonderfully the vector structure of the space […]. The fact that a triangle is half of a parallelogram has not prevented one to put emphasis for more than twenty centuries on a detailed study […] of triangles […]. One saw the triangle, but not the parallelogram which could have led to vectors.] (Choquet, 1964, p. 10) Continuing with the definitions of an affine transformation, a translation, a homothety, and a dilatation, there are some differences in the order in which these concepts are treated and in the importance that is given to them. Loyal to his choice of the axiom system, Choquet first defines a translation as an operation on a straight line, before extending this definition to the plane. Dieudonné first defines these transformations as mappings on the vector space, even before defining a straight line. The definition of a dilatation is in Dieudonné's book left for the exercises, which should not be a surprise, since a dilatation is either a translation or a homothety.

Entering the world of Euclidean geometry, the consequences of the different choices for the axiom system, seem, at first sight, to manifest themselves again more firmly. At this point, Choquet first mentions briefly that only basic and well-known properties of real numbers will be used. He then posits in an axiom the concept of distance as a structure on a straight line. Later, in his chapter on the axioms of the metric structure, he first states an axiom on when two straight lines are perpendicular, followed by the definition of perpendicular projection. Only then, the notion of distance is used to define the norm of a vector. This concept of norm, together with the perpendicular projection, finally leads to the definition of the scalar product of two vectors.

Dieudonné reverses this order more or less completely. Since he starts with a vector space as underlying structure of the Euclidean space, he disposes immediately of the scalar product of two vectors. Similar to Choquet, however, he restricts this scalar product as an operation on a straight line. Applying the scalar product, the definition of a norm of a vector and the distance between two points, follow immediately. Only then, the notion of two perpendicular vectors is defined, again using the scalar product of these two vectors.

In his chapter on the metric structure of Euclidean space, Choquet mentions that the axioms he presented allow to show that the concept of distance that he introduced is equivalent to the concept of distance that is associated to the scalar product (which is the way Dieudonné defines distance), summarized in a Theorem: Théorème 96.3. Pour tout point , on peut définir sur , de façon unique, une structure d'espace vectoriel d'origine , muni d'un produit scalaire, dont les variétés affines de dimension 1 et 2 sont les droites et plans de , et dont la distance associée au produit scalaire soit identique à la distance donnée sur . [Theorem 96.3. For any point , one can define on , in a unique way, a structure of a vector space with origin , equipped with a scalar product, whose affine subspaces of dimension 1 and 2 are the lines and planes of , and whose distance associated with the scalar product is identical to the distance given on .] (Choquet, 1964, p. 148) The treatment of the Euclidean transformations or isometries (i.e. translations, rotations, and reflections) is very similar in the books of Choquet and Dieudonné. Of special interest is that both authors first define a rotation, and then use the group of rotations to present a definition of an angle. As Choquet formulates it, a substantial part of geometry can be constructed without the use of angles: remarquons que nous avons pu construire commodément une grande partie de la géométrie sans parler jamais d'angles: La structure affine de Π, le théorème de Pythagore, la théorie des similitudes, ont été établis sans utiliser, ni angles, ni cas d'égalité des triangles. [note that we could conveniently construct a large part of geometry without ever talking about angles: The affine structure of Π, the Pythagorean theorem, the theory of similitudes, were established without using neither angles, nor similar triangles.] (Choquet, 1964, p. 97) Only in a later stage they write about trigonometry and trigonometric formulas and about measuring angles. Both Choquet and Dieudonné end their exposition on the "best" axiom system by looking beyond secondary school geometry by examining in the appendices some extra topics, such as angles from a different viewpoint, non-Euclidean geometry, and quaternions and rotations.

A settlement between Choquet and Dieudonné

Papy's didactic approach, as elaborated in his Mathématique Moderne (Papy, 1964(Papy, -1967)), soon enjoyed considerable prestige in circles of mathematics educators, particularly in France (see, e.g., [START_REF] Walusinski | La réforme est en acte[END_REF]. Also [START_REF] Dieudonné | Algèbre linéaire et géométrie élémentaire[END_REF] had praised "the remarkable and promising trials of our Belgian neighbors" (p. 17). Papy's approach to the teaching of geometry basically consisted of two stages. In the lower grades (12-15-year-olds), a system of (synthetic) axioms for affine plane geometry was gradually introduced, leading to the structure of the vector plane, an approach showing strong parallels with Choquet's. In the third year of secondary school (14-15-year-olds), this structure was equipped with an inner product, leading to the so-called Euclidean vector plane. At the beginning of the upper grades (15-18-year-olds) then followed what Papy called a "retournement psychologique" [psychological reversal]: The achieved goal of the first stage-the Euclidean vector space structure of the plane-was taken as a new and "unique" axiom for the second stage, i.e. the further development of geometry (of the plane and of higher dimensions) from a purely algebraic perspective.

At a meeting of the CIEAEM in Milano Marittima, Italy in 1965, and titled "The place of geometry in modern mathematical teaching", André Revuz tried to reconcile Choquet's and Dieudonné's points of view [START_REF] Revuz | Pour l'enseignement de la géométrie, la route est tracée[END_REF]. Revuz, who was esteemed by all parties, presented a statement about the role of geometry in the education of 12-to 18-year-olds, based on Papy's two-stages approach which was agreed upon by the CIEAEM members present. In the same year, on the occasion of an ICMI colloquium in Echternach, Luxembourg, this proposal was solemnly approved by Choquet and Dieudonné (De Bock & Vanpaemel, 2019).

In conclusion

Excesses of modern mathematics of the 1960s, such as the one described in this paper, have likely contributed to the current divide between the mathematics community and the mathematics education community, a divide that is unhealthy for both (Fried & Dreyfus, 2014). As concluded by [START_REF] Kilpatrick | The new math as an international phenomenon[END_REF], although modern mathematics is considered a failure, it did change mathematics education and the way it is treated by its stakeholders. Likewise, the books discussed in this paper were criticized [START_REF] Freudenthal | Review of Algèbre linéaire et géométrie élémentaire by J. Dieudonné[END_REF] and presumably not often used in secondary education, but they did stir the debate on geometry education in the 1960s and afterwards, geometry education did not return to the state it had before Dieudonné initiated the debate. Although the outcome of this debate on geometry education is in large likely due to the diplomatic qualities of Revuz, further research on the role of Papy's approach in this settlement could be of interest.

All translations were made by the authors.


Jenneke Krüger
email: j.h.j.kruger@uu.nl

Concurrence of two mathematics worlds in the Netherlands, 1600 and beyond

Keywords: History of mathematics education, mathematics for practitioners, engineering school, mathematics as school discipline, mathematics teachers

In the early 17th century at the University of Leiden, mathematics courses in the Liberal Arts and a mathematics course for surveyors and engineers in Dutch language existed peacefully next to each other, with some pathways between them. The reasons for this unique situation, the relation between the different mathematical courses and the characteristics of the Dutch language course are discussed. The design of the Dutch language course has some aspects in common with modern curricula. Two questions arise. Did this course influence the mathematics education in the next centuries? Was the Dutch language course a first step towards the development of mathematics as a school discipline? The second part of the paper attempts to find answers to these questions.

Introduction

Towards the end of the 16 th century two different worlds of mathematics and mathematics teaching may be distinguished in North-western Europe. One was the learned world of mathematicians at universities, at other institutes of learning, or working independently, whose object of study was the mathematics of the Greek mathematicians, their translators and commentators. It was a world of theory, in which an increasing number of ancient works became available for study, encouraging the development of mathematical theory. The language of communication was Latin, obligatory at all universities, the language of the men of learning. Mathematics teaching formed a part of the Liberal Arts, the general education as a foundation for the higher faculties. The students followed at least lectures on pure mathematics (number theory and Euclidean geometry), sometimes followed by mixed mathematics, subjects such as astronomy, mechanics, and cosmography.

The other world was the practical world of the common people, practitioners who didn't know Latin and who used some mathematics as a tool in their working life. Their grasp of mathematics might be extensive, but often it was very limited, without much theoretical background. In this practical world of surveyors, bookkeepers, building masters, military architects, traders, navigators, etc. there were as yet no teaching institutes. Private instruction by reckonmasters, self-instruction and learning at work were the main avenues to mathematical knowledge. The mathematics taught was primarily practical, without much attention to mathematical theory, confined to routines that seemed to work in a specific situation. This changed in Leiden around 1600, at the university, which was established in 1575 as the first university of the northern Netherlands. The mathematics lectures, in Latin, in the Liberal Arts faculty were taught by Rudolph Snellius, while from 1600 a Dutch course for surveyors and military engineers flourished, under responsibility of the University and taught by university professors. This last course became known as Duytsche Mathematique (Dutch Mathematics). The reasons for this unusual situation, some characteristics of and the relations between the two mathematics form the subject of the first section of this paper. Two questions arise. 1) Are influences of the characteristics of Duytsche Mathematique recognisable in mathematics education in the Netherlands during the 17 th and 18 th century?

2) Was the engineering course an isolated incident or a first step towards mathematics as a school discipline?

Following [START_REF] Chervel | L'histoire des disciplines scolaires: réflexions sur un domaine de recherche[END_REF][START_REF] Cardon-Quint | L'histoire des disciplines: Un champ de recherche en mutation[END_REF] a school discipline is taken as a subject taught in institutes, with specific teaching and learning goals, development of specific content partially independent of academic mathematical sciences, dedicated teaching methods, learning means, tests and examinations and taught by a body of professional subject teachers.

Mathematics teaching at Leiden University in the early 17th century, an innovation

The governors of the new Leiden University did their utmost to invite humanistic scholars with a good reputation to become professor in Leiden. As was the case in the older renowned universities, the theoretical framework for the faculties consisted of the works of Greek and Arabic authors, such as Aristotle, Euclid, Ptolemaeus, Hippocrates, Galenus, Avicenna and Averroës [START_REF] Otterspeer | Groepsportret met Dame I. Het bolwerk van de vrijheid[END_REF]. Justus Lipsius, Joseph Scaliger and Claudius Salmasius were some of the more famous scholars who came to Leiden, in return for a very good salary, free living and exemption of the obligation to give public lectures. Their names warranted quality of the university; they were expected to attract many students.

The first appointments were in the three main faculties: Theology, Law and Medicine, followed by Literature and Languages (classical and eastern) and Philosophy. Mathematics had a low priority; during the first years there was no professor for the undergraduate mathematics courses. In 1581 Rudolph Snellius (1546-1613) was appointed, a graduate in the Liberal Arts of the Calvinist University of Marburg. In Marburg he became an enthusiastic follower of Petrus Ramus (1515-1572), the French humanist and educational reformer. Ramus had proposed a shorter, wellstructured curriculum, which combined theory with practical exercises. For mathematics he proposed to teach only those parts of Euclid and other authors which were useful to the students and to combine theory and practice [START_REF] Wreede | A Humanist reshaping the mathematical sciences[END_REF].

Snellius was not the only follower of Ramus at Leiden University, moreover he favoured teaching a variety of authors, both the ancient and modern. Some of the authors he taught were Ramus, Valerius (Physica), Euclid (Elements, Optics), Maestlin (a follower of Copernicus) on cosmography and astronomy, and classical authors such as Aratus, Proclus and Plinius (on geography). Being a Ramist meant he would be inclined to value mathematics used for practices on its own merits.

Around 1605 Rudolph gave lectures on geometry, geography and optics, his son Willebrord assisted him with lectures on arithmetic and astronomy. Willebrord Snellius (1580-1626) was taught by his father, he also was a student of Ludolf van Ceulen (see below) and of Joseph Scaliger. He went on to become a highly praised and very productive mathematician, who succeeded his father in 1613 [START_REF] Wreede | A Humanist reshaping the mathematical sciences[END_REF].

Towards the end of the sixteenth century the commanders of the Dutch army, Maurits van Nassau (1567-1625) and Willem Lodewijk van Nassau (1560-1620), realized they would need many more mathematically trained engineers and surveyors to maintain their military advantage over the Spanish army. There were a few mathematically knowledgeable military engineers such as Adriaan Antonisz. and Samuel Crop and some practitioners such as Ludolf van Ceulen, who based their practices on sound theoretical mathematical knowledge, without knowledge of Latin, but they were few. Maurits, son of Willem van Oranje1 , had been a student at Leiden University, where he met Snellius and a fellow-student, Simon Stevin (1548-1620). Stevin, a good mathematician and engineer, became the tutor, quartermaster-general and principal advisor of Maurits. Together they convinced the curators of the University of Leiden to establish a separate course in Dutch language for surveyors and military engineers, the Duytsche Mathematique. It started officially in January 1600 with a curriculum prescribed by Stevin in the Instruction. Stevin designed an efficient curriculum, with the content, teaching methods and work forms prescribed. Theory preceding and alternating with practice; a method of study very much favoured by Stevin [START_REF] Krüger | A first mathematics curriculum: Stevin's Instruction for engineers (1600)[END_REF]. The course itself, the use of a written curriculum and the teaching language were an innovation.

The first professors who taught Duytsche Mathematique were Ludolf van Ceulen (1540-1610) and Simon Fransz. van Merwen (1548-1610). Simon Fransz. Van Merwen was a surveyor and burgomaster of Leiden. Ludolf van Ceulen was a respected mathematical practitioner, who taught mathematics and fencing and served on some committees advising the government. He didn't know any Latin but was very accomplished in geometry. He is best known for his calculation of the first 35 decimals of . Van Ceulen passed away in December 1610, whereupon his former student, assistant, surveyor and army engineer, Frans van Schooten, continued the lectures and the practice sessions in the field. At first as a temporary lecturer, from 1615 as professor Duytsche Mathematique [START_REF] Krüger | Lecture Notes for the First Dutch Course for Engineers[END_REF]. His extensive and well-illustrated lecture notes2 (BPL 1013) provide information on the content, structure, and didactic approach of the course. Van Schooten followed broadly the Instruction, with some adaptations and with deliberate didactics. In BPL 1013 calculations of roots, using decimal numbers is followed by some geometry and the principles and practice of surveying. This is followed by advanced surveying techniques, advanced geometry, calculations of volumes and wine gauging. Students who had come this far could take an exam to become an admitted surveyor, Rudolph Snellius had been an examiner on behalf of the university.

The second part of the lecture notes is about fortification, defined by Van Schooten as the art to build and defend fortifications as well as the art to attack and conquer fortified places.

In 1646 Frans van Schooten Sr. was succeeded by his eldest son, Frans van Schooten Jr., a very talented mathematician, who edited a new edition of Viète's work and translated Descartes' Geometry into Latin. He used the lecture notes of his father, but also gave lectures on logarithms, algebra, and sundials [START_REF] Dopper | A life of learning in Leiden[END_REF]. Though Frans Jr. was more a scholar than a practitioner, he maintained contacts with surveyors and engineers, even if he was not himself a surveyor and engineer as his father had been.

Ludolf van Ceulen, Simon Stevin, Willebrord Snellius and Frans van Schooten Sr. all served as examiners for the admission of surveyors in the province of Holland [START_REF] Krüger | Actoren en factoren achter het wiskundecurriculum sinds 1600[END_REF]. In the academic lectures the professors taught the theory of mathematics as developed by Greek and Arabian scholars and their interpreters, with Latin as teaching language. The Duytsche Mathematique on the other hand taught which mathematics to use and how to use it in specific practical situations, with Dutch as teaching language, in interaction with practitioners. In Leiden, during the first half of the 17 th century these two mathematics programmes, with different aims and content, existed peacefully together with links between them through the actors involved. Many factors, political, social, personal, were at the basis of this unusual situation.

Mathematics education during the 17 th and 18 th century

In 1681 the Duytsche Mathematique was abolished. It was reintroduced in 1701, as a theoretical university course, taught by a lecturer and it never regained the fame it had in the 17 th century. Until well into the 19 th century mathematics remained a rather unimportant part of the undergraduate program. However, the interest of university professors in technical applications increased. For example, civil architecture, hydraulics, steam engines, windmills, optics, navigation, and fortification were at one time or another subject of lectures at the universities of Leiden, Franeker and Utrecht [START_REF] Krüger | Actoren en factoren achter het wiskundecurriculum sinds 1600[END_REF].

In society the notion that mathematics was relevant for many crafts and professions gained acceptance. In primary schools there was a gradual improvement in the position and quality of arithmetic teaching. Virtually every village and town had at least one primary school, for which the local government was responsible. In the more ambitious towns mathematics formed part of the comparative examinations for a teaching post, a mathematics journal for teachers was published from 1754 (Krüger, 2019). Between primary education and university education the contours of on the one hand general education, on the other hand vocational education became visible.

General education as a preparation of boys for university took place in Latin schools and Illustrious schools. In so-called French schools general education prepared the sons and daughters of well-todo citizens for their position in society. Latin schools offered no mathematics or sciences, Illustrious schools offered a range of subjects in Latin, but some mathematical topics were taught in Dutch.

Examples are fortification and navigation at the Illustrious school in Amsterdam from 1743-1762; geometry, arithmetic, algebra, and fortification were taught at the Illustrious school in Deventer from 1690-1727. The mathematics taught in French schools depended on the headmaster-owner. In all these institutes the content was mainly determined by the teacher and was theoretical, though teaching applications. There was no formal curriculum for any type of school.

The perceived relevance of mathematics for many professions and crafts was reflected in the relative abundance of textbooks being printed and in the existence of many small private schools and teachers offering private mathematics lessons (geometry, navigation, fortification, bookkeeping, etc.). Naval colleges were established to teach mathematical skills to navigators and from the late 18 th century drawing schools and lessons in mathematics and drawing in orphanages became more common. Both in vocational training and in general education, mathematics teaching was haphazard, depending on initiatives and skills of individuals, with no written curriculum nor a system of examination. Neither was there a group of professional mathematics teachers (Krüger, 2019). However, from 1756 onwards the three institutes of the Foundation of Renswoude were comparable to the Duytsche Mathematique. In these institutes, in The Hague, Delft and Utrecht, talented orphans received a combination of professional training, based on mathematical subjects, and general education. From the start governors and mathematics teachers formulated a curriculum with subjects, aims, theory preceding practice and alternating with practice, examinations, textbooks, and other materials for the students. The first mathematics teachers had good contacts with many different practitioners, there were also contacts with professors at the Universities of Leiden and Utrecht [START_REF] Krüger | The power of mathematics education in the 18 th century[END_REF].

Unlike the situation in some other countries [START_REF] Karp | Dynamics of changes during the eighteenth century[END_REF] the Dutch government did not establish primary, secondary, or vocational institutes, until the end of the century (Krüger, 2019a).

Mathematics: a school discipline in general education

During the 18 th century, under the influence of French authors, mathematics gradually was considered as formative for the development of the mind, to learn to reason clearly and with use of logic. As a result, when around 1800 education became the responsibility of the national government, mathematics was on the agenda. Primary education was regulated first, in 1806; arithmetic became a compulsory subject. Moreover, all teachers had to pass exams, including arithmetic, to be allowed to teach and the inspectorate became rather important in the acceptance of modern ideas with regards to content and methods. From about 1820 the development of mathematics as a school discipline becomes noticeable: textbooks specifically for use in (extended) primary schools, specialized journals, teachers specializing in mathematics and authors discussing teaching methods (Krüger, 2019). The authors might be teaching at the Royal Military Academy, at French schools and other private schools, at naval schools and other similar institutes or they belonged to the group of primary school teachers in the highest rank, who had specialized in mathematics. Those who became proficient in mathematics could improve their position [START_REF] Smid | Dutch mathematics teachers, magazines and organizations[END_REF]. The regulation of secondary education had to wait until 1863. At least until then there existed a patchwork of private and council schools, extended primary schools and vocational institutes, all teaching some or quite a lot of mathematics, with textbooks written by some of the teachers.

From the start of the 19 th century mathematics was seen by regulators, inspectorate and others as belonging to general education, with a formative value in sharpening reasoning faculties. On the other hand, for many people the main aim of teaching mathematics was to prepare for crafts and professions. In their opinion the emphasis of teaching mathematics ought to be on mathematics used in professions and how to use this. Tensions between these aims surfaced occasionally, as in the case of Jacob de Gelder, a highly praised mathematician and teacher, author of many textbooks [START_REF] Beckers | Lagrange in the Netherlands. Dutch attempts to obtain rigor in the calculus[END_REF]. In 1819 he left the School for Artillery and Military Engineers after serious disagreement with the director, a military man, about the amount and level of mathematical theory which was taught (Krüger, 2019a). Another example is the appearance of an arithmetic journal aimed at crafts men and farmers, as the calculations at primary schools were deemed to be too exact and rigorous for daily practice (Krüger, 2019). Indeed, most pupils would enter the workforce after leaving primary school and learn a craft. The development of mathematics as a school discipline, visible in the first half of the 19 th century, received a strong impulse through the regulation of secondary education in 1863. For general education two new types of school were introduced, of which the Hogere Burger School3 (HBS) would become the most influential [START_REF] Krüger | Actoren en factoren achter het wiskundecurriculum sinds 1600[END_REF]. At the HBS the students were taught a range of subjects, with sciences, mathematics and modern languages taking up many hours in the timetable. The school had as one of its aims preparation for the Polytechnic School, also established in 1863, the successor of the ailing Academy for civil engineers (Krüger, 2019a). In the early years of the HBS mathematics included subjects such as mechanics, use of technology, cosmography, and line drawing, next to arithmetic, algebra, geometries, and trigonometry. The development of a group of professional mathematics teachers got a strong impulse through the HBS. Gradually topics such as mechanics and cosmology disappeared from the mathematics curriculum. From the last quarter of the 19 th century there were discussions about the content of the curriculum, the teaching methods, examinations and the merit of various textbooks. The teaching was theoretical, the use of mathematics in practice was delegated to the final years of Engineering Studies. At first, many mathematics teachers had a background in military or civil engineering, gradually most mathematics teachers had a degree in mathematics, but no experience with the use of mathematics in practices.

Discussion

Are influences of the characteristics of Duytsche Mathematique recognisable in mathematics education in the Netherlands during the 17 th and 18 th century?

As described in the second section, during the 17 th and 18 th century there was a growing awareness of the relevance of mathematics for numerous practices. Illustrious schools, small private schools and private teachers offered the teaching of mathematics to use in practices, or as part of general education. Mostly these programs or schools lasted at most a couple of years and details of the content are unknown. The only known example of a programme which showed some of the characteristics of the Duytsche Mathematique is the curriculum of the Foundation of Renswoude.

The three institutes of this Foundation resemble Duytsche Mathematique in the combination of theory and practice, the structure of the programme, the interaction with practitioners and the relative success; moreover, professors at the universities of Utrecht and Delft occasionally served as examiner for the Foundation in Utrecht and Delft [START_REF] Krüger | Actoren en factoren achter het wiskundecurriculum sinds 1600[END_REF].

Another sign of the awareness of relevance of mathematics for practices is the publication of numerous books on mathematical subjects, both the fundaments and the practices. Books on practices offered some mathematical theory; further research is necessary to determine the relevance and the depth of the treatment of mathematics in these books.

Was the Duytsche Mathematique an isolated incident or a first step in the evolution of mathematics as a school discipline?

The engineering course, the Duytsche Mathematique, at Leiden University was at the time innovative, due to the teaching language, and the way it was designed. A written curriculum, prescribing content, teaching methods, work forms, theory preceding practice and alternating with practice, is reminiscent of mathematics as the present-day school discipline. However, while the teaching method as seen in the manuscript by Frans van Schooten Sr. shows strong didactics, directed towards the needs of the students, the content was mostly aimed at the practices of surveying. Not all of it, there are geometry examples which do not seem relevant for surveying or engineers but are mathematically interesting. Also, there are indications that the practice somehow disappeared from the course, Frans van Schooten Jr., an excellent mathematician, seems to have concentrated on teaching and expanding the theory of the subject, with theoretical exercises about the practice of surveying, application of theory, as opposed to exercises in the field, using mathematics. One could say that Duytsche Mathematique showed some characteristics of a future school discipline, however it may also be seen as a forerunner of Engineering Studies.

The first example of a possible development of mathematics as a school discipline in the 18 th century is given by the Foundation of Renswoude. The programmes of the three institutes started coordinated, with some contact between the teachers in the three towns, with theory as the basis for practice. Interaction with practitioners was relevant for the programme, which could be described as a combination of general education with vocational training.

General education, with mathematics as a permanent factor, gained terrain from the start of the 19 th century, in the form of primary education. This was followed in the second half of the century and in the 20 th century by secondary education and the first years of vocational training. General education is theoretical, mathematics is about theory of the school subject, formulated in its own language and applied to theoretical examples. The use of mathematics, which mathematics to use and how to use it, is rarely or not at all experienced in general education and there is no interaction with professional users of mathematics.

Was the Duytsche Mathematique a first step in the evolution of the school discipline? Yes and no. It was more an example to follow than a first step; the Dutch society was not yet ready for more or less autonomous school subjects. A hundred years would pass before schoolteachers started to specialize in mathematics and another hundred years before a recognizable body of mathematics teachers became visible. They formulated opinions about the mathematics teaching programme, produced teaching materials and contributed to the formation of the school discipline, losing the interaction with professional users of mathematics in the process. Meanwhile the two worlds of mathematics from 1600 blurred and branched in many different ways, developing into a rich and varied universe of mathematics.

Willem van Oranje was the leader of the Dutch revolt against the Habsburg rulers, until his murder in 1584.

BPL 1013, University Library Leiden, special collection

Higher Burgher School, meant for the sons of middle-and upper-class citizens