Sci & Educ DOI 10.1007/s11191-015-9760-z

Negotiating the Boundaries Between Mathematics and Physics The Case of Late 1950s French Textbooks for Middle Schools Catherine Radtka1

 Springer Science+Business Media Dordrecht 2015

Abstract This paper examines physics and mathematics textbooks published in France at the end of the 1950s and at the beginning of the 1960s for children aged 11–15 years old. It argues that at this ‘‘middle school’’ level, textbooks contributed to shape cultural representations of both disciplines and their mutual boundaries through their contents and their material aspect. Further, this paper argues that far from presenting clearly delimited subjects, late 1950s textbooks offered possible connections between mathematics and physics. It highlights that such connections depended upon the type of schools the textbooks aimed at, at a time when educational organization still differentiated pupils of this age. It thus stresses how the audience and its projected aptitudes and needs, as well as the cultural teaching traditions of the teachers in charge, were inseparable from the diverse conceptions of mathematics and physics and their relationships promoted through textbooks of the time.

Education can be considered as a site for mathematics and physics’ interplay. For instance, Belhoste (1990) has shown that mathematicians involved in an important reform dealing with the French secondary instruction in 1902 considered geometry as a truly physical science and suggested evolutions of the curricula to promote their conceptions.1 Warwick (1992, 1993, 2003) and Navarro (2009, 2013) have brought to light the influence of the Cambridge pedagogical tradition of the ‘‘Mathematical Tripos’’ over physics in Britain: their focus on pedagogy has helped to understand how the production, reception and 1

This reform concerned specifically secondary instruction. It also witnessed numerous debates regarding competing conceptions of physics. The latter resulted in the introduction of practical exercises and practical works (travaux pratiques) in physics teaching in relationship with the adoption of a more experimental pedagogy. The context, debates and stakes of the reform regarding physics are the subject of a volume edited by Hulin (2000).

& Catherine Radtka [email protected] 1

Institut des Sciences de la Communication CNRS/Paris-Sorbonne/UPMC (ISCC), Paris, France

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transmission of new physics at the beginning of the twentieth century, such as relativity and quantum physics, were profoundly shaped by mathematical skills taught to undergraduate students. In this paper, I aim at exploring further the interplay of mathematics and physics in education by taking on a historical and cultural perspective: I study how mathematics and physics textbooks, through their content, their structure and their material aspect, influence cultural representations of both disciplines and their mutual boundaries. In order to do so, I focus on books published at the end of the 1950s and the beginning of the 1960s for 11- to 15-year-old children accommodated in the French educational system. The focus on textbooks for 11- to 15-year-old pupils is useful to examine cultural representations and images of both disciplines promoted towards wide audiences, an important part of which was not engaged in elite training or academic paths leading to scientific careers. Even though late 1950s education policy in France was driven by the aim of recruiting more scientists, technicians and engineers, effects of such a policy cannot indeed be reduced to the issue of future scientists’ training (Shapiro 2012). Rather, their analysis helps identifying and understanding various and socially diversified practices and conceptions of science (D’Enfert 2012a; Gispert and Schubring 2011; Rudolph 2002, 2005). In what follows, I take on the latter works to make the hypothesis that even though academic conceptions of scientific subjects are essential in the definition of disciplines’ identities, organization, aims, tools, and actors involved in science education for large audiences also contribute to such definitions. In other words, I wish to confirm the dependence of these disciplines’ identities and relationships on social and cultural parameters. In order to do so, I focus on a specific time period when the French educational system remained socially organized and offered different teaching to children with different social backgrounds, but encountered important changes. Indeed, during the decades following World War II, France witnessed a growth in general schooling and a raising up of school-leaving age which interfered with debates regarding mathematics and science teaching. After the war, discussions regarding education were dominated by the preparation of an important institutional reform: between 1947 and 1959 (when the Berthoin reform actually took place), 14 reform projects had been worked out to change the French system of education. All planned to widen access to secondary education. Even though the growth of the number of pupils engaged into postelementary education had not awaited an actual reform (Chapoulie 2010), it constituted after World War II a particularly acute problem for civil servants within the Ministry of Education and educators: emergency measures were taken in order to accommodate the growing number of pupils, new school subjects and options were developed, and renewed pedagogical methods were experimented (D’Enfert and Kahn 2010; Frank and Mignaval 2004). In such a context, the contents, methods and aims of mathematics and science teaching were also called into question, both for primary and secondary grades (D’Enfert 2010, 2012b; Gispert 2009; Guedj and Kahn 2010; Savaton 2010). Besides, those reforms provided for the merging of primary and secondary schooling at the middle school level.2 Distinct traditions of education were confronted, notably in the field of mathematics (D’Enfert 2012c). Hence, focusing on the late 1950s, when debates were particularly strong among educators and new textbooks series were issued further to the publication of 2

Until the 1959 reform, in France, the words ‘‘primary’’ and ‘‘secondary’’ did not refer to levels of education but to ‘‘orders’’: this situation, known as the ‘‘educational duality’’, established autonomous and parallel courses of study, which were more or less impenetrable (see Sect. 1.1). By calling the different types of schools which accommodated 11–15-year-old children the ‘‘middle school level’’ I follow Renaud D’Enfert’s introduction of a useful category for analysis.

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new national syllabi, retains a specific interest in order to study possibly divergent conceptions of relationships of mathematics and physics, and transformations at their mutual boundary.3 In the following, I start by examining the nature of science textbooks for pupils aged between 11 and 15, and I link it to their different audiences. This first section is based on the inventory of textbooks I have made thanks to the analysis of the annual supplement to the French Bibliography which was at the time published every new school year: this publication put together publishers’ catalogues; its use allows identifying textbooks series and their authors.4 Then, I turn to the analysis of the books themselves which I consider for their contents and for their material aspects: this brings up a material and visual differences between mathematics textbooks on the one hand and physics textbooks on the other hand that contributed to distinguish both disciplines. The next section examines more in detail the sharpness of the disciplinary demarcation and brings to light disciplinary intersections. I conclude by arguing that making sense of such intersections requires to take into account textbooks audiences and their authors’ conceptions.

1 Demarcating the Disciplines: Distinct Textbooks for Distinct School Subjects Mathematics and physics are distinct school subjects—and are taught with the help of distinct textbooks. Such a statement might seem straightforward, but it cannot be considered as self-evident: it results from specific decisions over the organization of education and from publishing choices. Nowadays in France, mathematics constitutes an independent school subject, while physics is taught together with chemistry [the name of the subject varies from ‘‘physical sciences’’, ‘‘physics and chemistry’’ (physique et chimie) to ‘‘physics–chemistry’’ (physique–chimie)].5 This situation can be dated back to decisions taken in the aftermath of the French revolution. As Bernadette Bensaude-Vincent et al. (2003, p. 54) indicate, ‘‘it is clear that for the legislators, the couple physics–chemistry had been constructed around the idea of experimental practice, well distinguished from the practice of observation peculiar to natural sciences, as well as from the practice of abstraction peculiar to mathematics’’. This provided for a tripartite division that lasted for a long time: in 1957–1958, the newly published syllabus for middle schools used both the terms ‘‘observational sciences’’ and ‘‘natural sciences’’.6 The tripartite division of scientific disciplines has nonetheless to be nuanced: together with regular calls to transcend too strong specializations (Hulin 2007, pp. 49–62), teachers’ 3

Such an interest is even stronger considering the relatively under-developed historiography on the second third of the twentieth century, when compared with the beginning of the Twentieth Century and 1970s reforms. In this paper, I follow a quite recent trend of scholarship which focuses on the period from the interwar years to the 1960s (see especially works by R. D’Enfert, P. Kahn and H. Gispert quoted above).

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Bibliographie de la France, supple´ment « rentre´e des classes » , Cercle de la Librairie (1958–1961).

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The hyphen is not used systematically.

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‘‘Observational science’’ refer to the syllabus for cours comple´mentaires, while ‘‘natural science’’ refers to colle`ges and lyce´es (see below for the differences regarding those types of schools), but the contents were roughly similar to each other. The official instructions are presented in the following texts: Arreˆte´ du 25 septembre 1957, Bulletin Officiel de l’Education Nationale [BOEN] n 35, 3 octobre 1957, pp. 2939-2941; Circulaire du 20 octobre 1957, BOEN n 42, 21 novembre 1957, p. 3429; Arreˆte´ du 10 octobre 1958, BOEN n 38, 23 octobre 1958, pp. 3061–65.

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Technical Lycées & Collèges Term Term Cours Complémentaires 3e (troisième)

1re

2

de

2de

3

e

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4e

4e

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classical

4e (quatrième)

End-of-studies end-of-studies 2

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end-of-studies 1 6 (sixième) Elementary School Middle class 2 (Cours moyen 2e année) re

modern 7e 8e

Middle class 1 (Cours moyen 1 année) e

Elementary class 1 (Cours élémentaire 2 année)

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Elementary class 1 (Cours élémentaire 1re année)

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1 year of primary school (Cours préparatoire)

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Fig. 1 French system of education in the late 1950s. On the diagram, the streams’ width roughly represents pupils’ numbers. I have not represented the vocational centres which accommodated a significant part of students who had completed the end-of-studies years. Before the 1959 Berthoin Reform, the elementary school, end-of-studies forms and cours comple´mentaires depended all on the primary order of instruction. The reform renamed the cours comple´mentaires ‘‘colle`ges d’enseignement ge´ne´ral’’ and united them with the secondary order of instruction. They delivered (before and after the reform) the same kind of short teaching. Until the reform, the secondary order of instruction proposed long and short, as well as classical and modern, training in colle`ges and lyce´es. Among the colle`ges, one type named ‘‘colle`ges modernes’’ offered only short training; other colle`ges offered the same teaching as lyce´es (short, long, classical and modern streams), but lyce´es were (usually) of better reputation. The technical schools (called colle`ges techniques) depended on a third (and more recent) order of instruction. The diagram is adapted from Frank and Mignaval (2004, p. 39)

training and curricula organization over time also account for proximities, which were stronger in the so-called primary order of instruction which in the late 1950s accommodated a large part of 11- to 15-year-old pupils.7 Another important part of pupils was accommodated in secondary schools. There, teachers and pedagogical practices and schooling organization were quite different from their counterparts in the primary order of instruction. Before I examine textbooks published for children of this age in physics and in mathematics, it is thus necessary to describe the late 1950s educational system in order to understand how science education was shaped and for what kind of audiences.

1.1 Teachers, Pupils, and Textbooks at the Middle School Level in the Late 1950s: The Encounter of Two Distinct Traditions In France, the school-leaving age was 14 until 1959, when it was raised to 16 by the Berthoin reform. Elementary education depended on the primary order of instruction and accommodated children up to (roughly) 11 years old. Thus, after completion of elementary education, an important part of pupils who were not expected to study beyond the compulsory school-leaving age had to pursue their studies for some more years: they went to end-of-studies forms (classes de fin d’e´tudes) and were taught by primary school teachers. Others joined streams where they could pursue their studies for 4 or 7 years. Such post7

See above footnote 3.

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elementary studies were delivered by the primary order of instruction and by the secondary order of instruction.8 Secondary schools were called lyce´es and colle`ges.9 Primary forms for 11–15 years old were called cours comple´mentaires (i.e. complementary courses).10 Differences between primary and secondary schools that accommodated 11- to 15-yearold children were important: they were first of administrative nature, as cours comple´mentaires on the one hand and colle`ges and lyce´es on the other hand did not depend on the same department within the French ministry of education. They were also of educational nature, as teachers had not been trained in the same way, and curricula were different. They were also sociological, as these types of schools did not traditionally accommodate the same children. This educational system, largely shaped by its strong division between two distinct and independent ‘‘orders’’, was an inheritance of an organization designed in the nineteenth century to match the projected social destiny of the children to train. Historically, secondary instruction has been mostly accommodating children of the social elite. It led to the most prestigious training institutions, to the highest positions within society, and to academic professions. It provided both short and long teaching. At the beginning of the twentieth century, secondary instruction had been divided into two cycles: some pupils were accommodated in short streams which lasted four years and ended with the first cycle; others kept studying in the second cycle in order to prepare the Baccalaure´at which opened the path towards university. Until the 1960s, the most prestigious streams of that order delivered an education dominated by the study of Latin, Greek, and classical humanities. However, modern streams which included more science attracted progressively more and more students and gained a better reputation.11 School subjects were modelled on academic disciplines; the more so because teachers had been trained at the university. Regarding scientific subjects, the sharpest boundary was drawn between mathematics and other scientific subjects. This situation was a legacy of earlier organization: from 1840 until 1869, prospective science teachers had had indeed the possibility to sit a prestigious exam that would allow them to teach in the lyce´es. This examination, called Agre´gation, would lead the laureate to be awarded the title of Agre´ge´ either in mathematics or in the triad physics–chemistry–natural sciences. Even once a specific Agre´gation in natural sciences had been introduced, natural science teaching in lyce´es remained frequently in the physics teacher’s charge (Hulin 2007, pp. 54–55). Paralleling such a situation, until 1941 the inspectorate (Inspection ge´ne´rale) of physical sciences was at the same time in charge of natural sciences (Savaton 2010, p. 102; Larue

8

A third—technical—order also accommodated some children in technical schools. It concerned a minority of students.

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The lyce´es were created in 1802 in order to form the future national elite: they were located in big cities and were maintained by the State. Some of these schools resulted of the transformation of older establishments, called colle`ges, which existed in the Old Regime. Not all colle`ges were transformed into lyce´es and cities could, at the beginning of the nineteenth Century, chose to maintain these establishments. Some delivered the same kind of instruction as lyce´es, others delivered shorter training, but all depended on the secondary order of instruction. The lyce´es remained the most prestigious establishments. At the end of the 1950 s, both structures still existed (even though their number, the way they were maintained, and the teaching they offered had changed), and the difference of reputation remained. For the history of lyce´es, see Caspard et al. (2005).

10

See above Fig. 1.

11

Modern streams also included the study of foreign living languages, and did not require the study of Latin.

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2002).12 Besides, physics–chemistry was only taught to children accommodated in short modern streams: others would start studying this subject only in the second cycle of colle`ges and lyce´es (i.e. the last 3 years of secondary instruction). Within primary instruction, the situation was altogether different. This order had been designed to train children of the people and to prepare them to enter into working life. It could nonetheless lead some children to obtain socially important positions: the primary order provided indeed for the training of its own teachers and for intermediary professions as it offered post-elementary education. A hierarchy also existed among teachers of that order. All future teachers started their training in e´coles normales where they received multidisciplinary education: they were expected to train all school subjects at an elementary level and in cours comple´mentaires. Some of them also received a more advanced education in Ecoles normales supe´rieures primaires at Saint-Cloud for future male teachers and Fontenay for future female teachers. Students of these schools were considered as the elite of the primary order; they were trained in order to teach in e´coles normales or in higher schools called e´coles primaires supe´rieures. During the first half of the twentieth century, the training they received was more specialized than the standard one: it was subdivided between arts and literature on the one hand, and science on the other hand. From 1921 onwards, the specialization has been pushed further: future ‘‘professors’’ (their official title) could get a specialization either in mathematics (which became mathematical and physical sciences in 1928), in physical, chemical and natural sciences, or in applied sciences. As in the secondary order, the triad physics–chemistry–natural sciences existed, but the division between mathematics and physics was softened: in the primary order, some teachers were ‘‘bivalent’’ math-physics and taught both subjects. The division of the educational system between primary and secondary order remained theoretically very structuring up to the 1950s. However, in the interwar period, measures taken by the minister of National Education Jean Zay tended to bring different forms for the same ages closer to each other.13 Then, a reform undertaken by the Vichy government transformed e´coles primaires supe´rieures into colle`ges modernes in 1940. Even though they kept providing the same kind of teaching to their pupils, it made colle`ges modernes and their teachers dependent on secondary instruction and contributed, quite unexpectedly for its promoters, to bring closer both orders (in the late 1950s, colle`ges modernes provided the majority of short modern training within secondary instruction). Cours comple´mentaires remained, for their part, in the jurisdiction of primary instruction. After World War II, they evolved and their curricula became more and more similar to first cycle of secondary schools. The 1959 Berthoin reform transformed the cours comple´mentaires into colle`ges d’enseignement ge´ne´ral and united them with the secondary order of instruction. However, before and after the reform they delivered the same kind of teaching, and publishers did not modify recent textbooks after the reform: they finished off the issuing of the textbooks series that followed the publication of the 1957 syllabi and, usually, modified the title from ‘‘cours comple´mentaires’’ into ‘‘colle`ges d’enseignement ge´ne´ral’’.14 For 12 Created in 1802, the Inspection ge´ne´rale had had until 1968 clear missions: its members inspected classrooms, presided over juries of teachers’ recruitment and had a real pedagogical authority. Over time, the corporation had grown up and had become more and more organized according to school subjects. Since 1968, its missions have profoundly changed (Rioux 2002). 13 At the time, the names ‘‘primaire’’ and ‘‘secondaire’’ were officially replaced by ‘‘premier degre´’’ and ‘‘second degre´’’. 14 Such a change most of the time took place for the books issued in 1961 which results in series including textbooks for the first years of cours comple´mentaires (issued up to 1960 and including one, two or three books according to the rhythm of publication) and then textbooks for the last years of colle`ges

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the sake of clarity and to underline the primary tradition of teaching in cours comple´mentaires and then colle`ges d’enseignement ge´ne´ral, I only use cours comple´mentaires in the article. In the late 1950s–beginning of the 1960s, the situation was thus marked by the legacy of distinct traditions, embedded in administrative, cultural and sociological differences, and, at the same time, by a more recent tendency which made curricula within cours comple´mentaires and first years of lyce´es and colle`ges closer to each other. Late 1950s textbooks reflected this double trend: some were specifically aimed at certain types of schools and thus fragmented the audience according to schooling organization; others were aimed at all pupils accommodated in all types of schools. Yet, all these textbooks shared some common points particularly in the way they contributed to draw disciplinary boundaries. Indeed, all textbooks, including those aiming at cours comple´mentaires, mirrored academic separations: more precisely, different series were published for mathematics, for natural sciences and for physics–chemistry (the latter were intended only for pupils of cours comple´mentaires and for pupils of short modern streams of colle`ges and lyce´es). This paper accordingly focuses on the textbooks issued for pupils receiving short training15: all together these pupils amounted in the late 1950s to almost the half of new generations.

1.2 Distinct Textbook Series in Mathematics and Physics for Middle Schools Pupils As previous comments indicate, this paper deals with mathematics textbooks on the one hand and the physics section of physics–chemistry textbooks on the other hand16: there were no textbooks, and even no series, which would have dealt with all the scientific disciplines (including mathematics).17 Such a situation reflected publishing choices which favoured academic division and contrasted in particular with the multidisciplinary culture of teachers of cours comple´mentaires. What is more, books did not only define the subjects’ limits because they materialized them in distinct objects; they also did so because their authors were not the same. Not a single author contributed at the same time to one of the 24 textbooks series published by 20 publishing houses in mathematics and to one of the

Footnote 14 continued d’enseignement ge´ne´ral. Only the publisher Belin reissued (only in 1963) a textbook for the sixie`me form following ‘‘the united syllabus of 1960’’. 15

It is important to note that even so these pupils might have received the same books in mathematics as pupils accommodated in long streams. 16 In order to lighten the phrasing, I use from now on ‘‘physics textbooks’’ instead of ‘‘physics-chemistry textbooks’’: late 1950 s textbooks usually included separated sections devoted to physics and chemistry. The series published by Masson under the name of Maxime Joyal featured separated books for physics and for chemistry and was, as such, an exception, but it strengthened the disciplinary division (Delattre and Boue´ 1961a, b, c, d). 17 Following the official syllabi, textbooks for Classes de fin d’e´tudes were slightly different: distinct textbooks existed for mathematics on the one hand, and for ‘‘applied sciences’’ on the other hand (which included notions in natural sciences). However, they maintained a sharp boundary between mathematics and other sciences. In another period, one textbook series which included mathematics textbooks within other science textbooks had been published: entitled the ‘‘Collection scientifique’’, it had been edited by Albert Chaˆtelet and published by Baillie`re in the 1930 s.

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Fig. 2 A. Payan, P. Chilotti, F. Boyet. (1962) Physique et chimie. Classe de 3e enseignement court et colle`ges d’enseignement ge´ne´ral. (2nd edition). Paris: Librairie Armand Colin (pp. 90–91). This section of exercises is taken from a chapter that lasts for 6 pages (chapter 19 ‘‘Conservation de l’e´nergie’’) ( Armand Colin 1960)

12 textbooks series published by 10 publishing houses in physics that are studied in the current article.18 In France, science textbooks have usually been written by authors belonging to the national education system. In the late 1950s, most of the authors were teachers who had been trained and who taught in the secondary order of instruction (and among which, most had passed the prestigious Agre´gation). Some were general inspectors (inspecteurs ge´ne´raux) which meant that, after a successful teaching carrier, they had joined the corporation of the Inspection ge´ne´rale.19 Even for textbooks aiming specifically at children in cours comple´mentaires, most authors thus came from the secondary tradition and had received university training in a specific subject. Such a situation could account for the apparently strict division between mathematics and physics authors. However, some authors as Albert Lermusiaux and Marcel He´meret, who wrote the sole mathematics textbooks series exclusively dedicated to cours comple´mentaires pupils, came from the primary order. Both were elite members of this order as their ‘‘title’’ of formers students of the Ecole normale supe´rieure de Saint-Cloud put forward in their textbooks indicated (He´meret and Lermusiaux 1958, 1959, 1960) and had received, as such, a multidisciplinary training. Others, like Andre´ Godier, who was presented in his physics textbooks as a former e´cole normale professor and inspector of primary instruction (Godier 18 These figures are given for information purpose only. They should not be considered as irrevocable ones: they depend on my choices to focus on children accommodated in cours comple´mentaires and short streams of colle`ges and lyce´es in the late 1950s, on the sources used to identify the textbooks series (the supplement ‘‘Rentre´e des classes’’ of the Bibliographie de la France published by the Cercle de la Librairie for the years 1958–1961) and on the categories used by the publishers’ catalogues to organize the textbooks production according to their audience. 19

See above, footnote 16.

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et al. 1960, 1961a, b, c), also belonged to the primary order, but he had followed, at some point in his training, university courses.20 Despite the multidisciplinary background of these authors, none of them participated simultaneously to the writing of mathematics and physics textbooks published in the late 1950s for 11- to 15-year-old pupils, even when they contributed to both disciplines textbooks at elementary level.21 Thus, at this level, distinct series of textbooks materialized a division between mathematics and physics and suggested that distinct authors were required to tackle these different subjects. For secondary schools pupils, books hence reinforced their everyday experience as mathematics and physics were taught by different teachers; for primary schools pupils, however, they were at variance with the image given by their teacher in charge of both mathematics and physics (and chemistry). The multidisciplinary tradition of primary schools was thus modified by the books the children read, even when their authors came from this tradition and had crossed boundaries between disciplines in their own training.

1.3 Different Styles for Different Disciplines The division between mathematics and physics was taken further by a material feature of the textbooks: all mathematics textbooks included numerous exercises and problems which constituted an important component of the chapters or lessons. By contrast, physics textbooks devoted less space and importance to exercises. For instance, the series published by Masson under Maxime Joyal’s edition usually dedicated a single page of exercises per chapter, and within the textbook for the troisie`me form (the fourth year of cours comple´mentaire, colle`ges and lyce´es), some chapters did even not include exercises (Delattre and Boue´ 1961a, c). In the physics series published by Hachette, the number of exercises was mentioned on the front page (Legreneur and Peyraud 1960, 1961), but they were typeset in two columns and in a reduced size font, thus introducing a clear visual change with regard to the lesson typeset in plain justification. Similar graphical choices also directed the visual aspect of the series published by Armand Colin (Payan et al. 1960, 1961). In this series, the space devoted to exercises rarely exceeded one page per chapter.22 In the series published by Nathan, an even more significant graphical choice had been taken: exercises were taken out from the ‘‘lessons’’ and grouped by sets inserted in between (Godier et al. 1960, 1961a, b, c). At the end of the lesson, a simple reference was made to corresponding exercises. Within mathematics textbooks, exercises belonged to the lessons whatever the graphical choices. For instance, the series entitled Cours de mathe´matiques Lebosse´ et He´mery after the names of its two authors presented exercises that were typeset in a reduced size 20 Andre´ Godier’s carrier path is actually remarkable as it calls into question the apparently strict separation between primary and secondary orders of instruction in the interwar years. Indeed, Godier started his teacher training as many other primary schools teachers in an e´cole normale (Saint-Loˆ from 1917 to 1920). He then pursued his training during an additional fourth year, and after his national service, he taught in a primary school and passed the examination in mathematics that allowed him to teach in e´coles normales. Afterwards, he was appointed to the e´cole normale de Saint-Loˆ. In parallel to his teaching, he studied at the university and succeeded in a bachelor’s degree in science, with a specialization in physics in 1933, which he ˆ me d’e´tudes supe´rieures’’ in physical sciences in 1934. He thus entered university completed with a ‘‘diplo after a primary order training and received training both in mathematics and in physics. These details come from his carrier records, archived under F1728308 in the French national archives. 21 This was, for instance, the case of Andre´ Godier wrote science and mathematics textbooks for elementary forms (Godier et al. 1958, a, b), and only wrote physics textbooks for later forms (Godier et al. 1960, 1961a, b, c). 22

See Fig. 2.

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Fig. 3 M. Monge, M. Guincham. (1958). La Classe de Mathe´matiques. 5e classique et moderne. Paris: Belin (pp. 14–17). These pages are taken from the 10-pages long chapter ‘‘decimal numeral system’’. Following a ‘‘reading’’, they include a section of ‘‘practical works’’ and ‘‘exercises’’. Besides, some ‘‘examples’’ and ‘‘exercises’’ followed by their ‘‘solutions’’ are included within the lesson itself ( Belin 1958)

(Lebosse´ and He´mery 1958a, b, 1960a, b, c, 1961).23 Their number and their place— following ‘‘practical works’’ (travaux pratiques)—indicated that such a choice did not parallel a lesser importance; it rather helped to include more exercises within a given space. In the series named after its editor Paul Dubreil, the page layout also broke with the visual aspect of the lesson.24 Yet it resulted in highlighting exercises by printing them on a coloured—rather than white—page (Brailly-Marchand and Fouche´ 1961). More generally, exercises appeared at the core of mathematical practice as they were in continuity with the worked examples and the type of discussion that was featured in the lesson per se. In addition to exercises, a strong emphasis was also put within mathematics textbooks on the logical organization of knowledge. French mathematics textbooks from the late 1950s followed a path which progressively introduced more abstract concepts and a greater use of deductive reasoning. The textbook series published under the name of Joseph Marvillet by Armand Colin even suggested by its subtitles that the true study of mathematics started in quatrie`me (the third form of cours comple´mentaires, colle`ges and lyce´es), while previous books only dealt with an ‘‘introduction’’ (initiation).25 Such gradation in abstraction was a shared feature of the different textbooks. As reasoning, demonstration and the logical organization of mathematics were commonly highlighted by

23

For another example, see Fig. 3.

24

The ‘‘collection Paul Dubreil’’ published by Vuibert included four books, aimed at lyce´es and colle`ges d’enseignement ge´ne´ral. Its editor was the mathematician Paul Dubreil who was then professor at the Faculty of Sciences in Paris. He had been appointed to the chair of arithmetic and number theory in 1954. All the textbooks of the series had their own authors. The analysis of the books contents does not suffice to know the part Paul Dubreil took in their edition. In each textbook, Paul Dubreil signed the foreword in which he highlighted the approach chosen by the authors to introduce their subject. 25 The Cours J. Marvillet published by the Librairie Armand Colin included five books. The first two books, for the sixie`me and cinquie`me forms of all types of schools, were entitled ‘‘initiation aux mathe´matiques’’ (first editions respectively issued in 1958 and 1959). The next three books (one for the quatrie`me form of all types of schools, one for the troisie`me form of cours comple´mentaires and another one for the troisie`me form of lyce´es and colle`ges) were entitled ‘‘mathe´matiques’’ (first editions respectively issued in 1960 and 1961 for both troisie`me books).

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Fig. 4 R. Maillard, R. Cahen, E. Caralp. (1960). Cours de mathe´matiques R. Maillard, 3e. Paris: Hachette (pp. 116–117) ( Hachette 1960)

the authors in the forewords, all books’ layouts (with more or less sophisticated means) notified the reader of the diverse nature of successive sections of the text.26 By contrast, within physics textbooks, the lessons’ core consisted in descriptive texts, whether they were used in order to describe a physical situation, an observation or everyday experience, or experiments. With regard to experiments, the rhetoric tended to make the reader a virtual witness of the work: the narrative started with general observations and progressively introduced scientific concepts by following an inductive reasoning. A strong emphasis was also put on the description, by words and pictures, of scientific instruments: their general aspect, their features and their proper use were described at length. For instance, within the physics textbook for quatrie`me of the Joyal series, the 21st chapter entitled ‘‘Temperature—Thermometer’’ devoted 4 pages out of 7 to the description of the mercury-in-glass thermometer and other ‘‘common thermometers’’ among which were more precisely presented the alcohol thermometer, the minima–maxima thermometer, the medical thermometer, the recording thermometer, and were mentioned metallic thermometers, resistance thermometers and optical pyrometers (Delattre and Boue´ 1961a).27 Besides, these descriptions followed the introduction of the notion of temperature, which was presented as the result one might read when he uses a specific device constructed by associating the phenomenon of thermal expansion and quantification. Such a construction of notions which put scientific instruments at the core of physical practice and knowledge was not limited to thermometry and to this specific textbook series. It was a more general pattern followed within all textbooks for quatrie`me where the notions of weight, forces, mass, pressure, and heat were introduced. Within textbooks for troisie`me, the pattern was taken further and the emphasis shifted from basic physical instruments towards more complex devices as the notions of energy, electricity and electromagnetism were studied. There, descriptions and images of engines, accumulators, 26

See Fig. 4 for an example.

27

See also Fig. 5.

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Fig. 5 A.-R. Delattre, R. Boue´. (1961) Physique 4e Collection M. Joyal, Colle`ges d’enseignement ge´ne´ral. Paris: Masson (pp. 168–169) ( Masson 1961)

dams, compasses and electromagnets were in continuity with descriptions and images of weighing scales, barometers and thermometers from the previous-year textbook. As a result, physics textbooks appeared much more illustrated than mathematics textbooks and made a more important use of photographs. An overall look at late 1950s mathematics and physics textbooks thus induced differences in the status given to exercises, in modes of reasoning (deductive vs. inductive), in the type of text (analytical vs. descriptive), and in visual aspects (more or less illustration, differences of layouts) which gave distinct identities to physics and mathematics. However, when the analysis is conducted on a smaller scale and on some specific chapters, boundaries between both disciplines do no longer appear so sharp.

2 Blurring the Boundaries: Disciplinary Intersections and Renewed Pedagogy Pages 164–165 and 182–183 from La Classe de mathe´matiques for sixie`me forms (that is first year of cours comple´mentaires or colle`ges and lyce´es) did not really fit within the description of mathematics layout given in Sect. 1: these pages included relatively long descriptive texts, no exercises, and were illustrated thanks to the use of diagrams, but also skillful drawings and even photographs (Monge and Guincham 1958a).28 They were visually close to physics textbooks’ pages such as the one reproduced in Fig. 4. Besides, they dealt with weight and time measurements, both subjects that could have found, and actually found to a certain extent, a place in physics textbooks (weight measurements were treated in books for quatrie`me forms). Even though measurement of time was only studied within mathematics textbooks, its closeness to physics appeared through the approach 28

See Fig. 6.

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Fig. 6 M. Monge, M. Guincham (1958). La Classe de Mathe´matiques, 6e classique et moderne. Paris: Belin. (pp. 164–165 (left) and pp. 182–183 (right)) ( Belin 1958)

adopted: as well as in physics textbooks, notions were progressively introduced thanks to observations of real-life experience, followed by descriptions of experiments or of instruments use and discussion of their results. These few pages were no exception. Thus, they cannot be considered as mere illustration or window on physics for the sake of interdisciplinarity: all mathematics textbooks series of the late 1950s similarly included entire chapters dedicated to very physical subjects and notions such as specific weight or uniform motion. Such contents fostered a conception of mathematics which included its applicability to real-world situations.

2.1 Quantification and Mathematical Concepts Applied to Real-World Problems: An Instance of Physics–Mathematics Interplay Within Mathematics Textbooks According to the national syllabi for which new versions appeared from 1957 onwards, textbooks for the sixie`me and cinquie`me forms had to deal with physical quantities and measurement units as well as with astronomy.29 Such contents gave an opportunity to introduce the metric system and to study astronomical phenomena from a geometric standpoint. However, in corresponding chapters, authors did not restrict themselves to the introduction of units and the handling of their conversions or to the picturing of an astronomical situation only preceding a very geometric treatment of—for instance—the position of noticeable stars in the sky. The textbooks series edited by the Inspecteur ge´ne´ral Roland Maillard provides more remarkable situations at the intersection of physics and mathematics.30 Taking up a postwar pedagogical innovation, all the chapters of the Cours R. Maillard started with

29 The presence of astronomy within mathematics can be seen as legacy of a nineteenth century consensus between mathematicians and physicists (Atten 1996). 30 Roland Maillard was a former student of the E´cole Normale Supe´rieure (Ulm) which trained elite teachers of the secondary order of instruction. In the late 1950s, he had joined the Inspection ge´ne´rale after a successful carrier as mathematics teacher in lyce´es. He had authored many textbooks for the publishing house Hachette before obtaining a textbooks series named after him. For this series, he remained an author and was helped by two mathematics teachers in Parisian lyce´es, Raymond Cahen and Euge`ne Caralp.

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‘‘practical works’’ (travaux pratiques).31 In the case of the chapter dealing with weight, the practical work required the use of different weighing scales and invited the teacher to use the equipment of the ‘‘Physics cabinet’’ (Maillard 1958, p. 191). It even went further in the blurring of a boundary with physics as the authors seized this opportunity to introduce physical definitions through the instruments used to measure identified physical quantities. Other chapters dealing with physical notions also devoted great space to the description of instruments as weighting scales, clocks, measuring tapes or surveying devices, and detailed what the proper uses of these instruments were. They explained the physical principle which lied behind the making up of instruments; they described their aspect which was also illustrated thanks to sketches, drawings and, remarkably enough, photographs; they also detailed what qualities were looked after depending on the type of use of the instruments (for instance sensitivity or reliability). As a result, sections of such chapters appeared quite similar to contemporary physics textbooks the pupils might use later on if they were accommodated in short modern streams of lyce´es and colle`ges or in cours comple´mentaires (see above Sect. 1.2.; see also Radtka 2013). Was there even any difference with physics textbooks? In the case of the Cours R. Maillard, the answer to this question ought to be positive for two reasons: first, because exercises the pupils had to solve clearly limited their scope to numbers and units manipulations and second, because at some point mathematics stepped back. Indeed, when it came to the difference between weight (poids in French) and mass (masse), the Cours R. Maillard merely made a short comment and left the explanations to the physics lesson (Maillard 1958, p. 193). More generally, short comments or the idea that some thinks ought to be ‘‘accepted’’ (admis) without further explanations contributed also to limit the scope of the questions the mathematics teacher could address in his lesson or during exercises. For instance, the mention of Archimedes principle within the sixie`me textbook of the Collection Paul Dubreil only served as a way to give some context to an exercise of ‘‘fast calculation’’ (Brailly-Marchand and Fouche´ 1961, p. 195). Within the latter textbook, authors also explicitly stated the idea that quantification was a requisite for reasoning and thus suggested that their interest towards real-life or physical situations constituted an instance of mathematics knowledge application: ‘‘We can see in everyday life that some moving bodies go faster than others, or that the same body can go more quickly at some point and less quickly at another, but these are usually only confused feelings. In order to be able to reason, it is necessary to start by observing a moving body which seems to move steadily and to try to express this feeling with numbers’’ (Brailly-Marchand and Fouche´ 1961, p. 206).

A similar idea resulted from the way geometry was introduced in the same book. It highlighted that geometrical notions were required to master the geography curriculum the very same year,32 and thus justified their treatment early in the book (Brailly-Marchand and Fouche´ 1961, p. 40). It became clear that geography (in a broad sense) constituted a field for the application of geometry as three photographs which illustrated the corresponding pages were captioned ‘‘Geometry applied to modern architecture. An example: the town of Royan’’ (Brailly-Marchand and Fouche´ 1961, p. 44). Contrary to physics textbooks, the use of photographs was quite unusual in mathematics textbooks, but they could foster the idea that mathematics was useful to understand physical and real-life 31

I comment on the pedagogy of ‘‘practical works’’ in Sect. 2.2.

32

In France, geography has traditionally been linked with history at school. However it was also considered as a subject connected to sciences, especially physics and geology; see (Hulin 2007, pp. 49–62).

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situations or that their applications encompassed their problems. Their presence could thus contribute to blur the boundaries between both disciplines by introducing the idea of a physical use of mathematical knowledge. However, late 1950s mathematics textbooks were not limited to that kind of interplay between mathematics and physics: they also made use of real and physical objects and situations in order to introduce mathematical knowledge.

2.2 ‘‘Practical Works’’ Within Mathematics: Physical Problems and Methods as Ways to Introduce Mathematical Knowledge In accordance with new versions of the syllabi that had been published from 1957 onwards, all late 1950s mathematics textbooks used ‘‘practical works’’ as a method to introduce mathematical knowledge, which reflected a relative consensus towards the use of ‘‘active’’ pedagogical methods (D’Enfert 2010). As Monge and Guincham (1958b, Avertissement) commented upon, ‘‘the official introduction of practical works has allowed us to develop this [active] method further in our teaching’’. This pedagogic innovation, which could actually be understood in different ways and stood for different kinds of activities, was generally praised by textbooks authors or editors in the forewords as a means to present mathematics in a lively and attractive way. However, taking into account such a method in order to write and organize a textbook could have remarkable consequences regarding the identity of mathematics by suggesting that a similar thinking as in physics could also apply in mathematics.33 Besides works that suggested using concrete objects, sometimes even manufactured by pupils in their manual works classes (He´meret and Lermusiaux 1958), or making use of everyday life experience in order to introduce mathematical objects and concepts as geometrical figures, ‘‘practical works’’ were also a means to conjecture mathematical relationships or proprieties. For instance, the volume of the Cours R. Maillard dedicated to troisie`me forms introduced new notions, as had the previous volumes, with ‘‘practical works’’. In this case, the chapter did not deal with physical quantities, or measurement units or astronomy, but with square roots, ratio and proportion, algebraic manipulations, Thales’ theorem, coordinates, the function y = ax ? b, and so on (Maillard 1960). Considering such notions, ‘‘practical works’’ did not necessarily consist in manipulation of concrete numbers and physical situation. For instance, the opening ‘‘practical work’’ introduced the first theorem of the book which stated that: ‘‘the difference of squares of two consecutive integers equals: 1 the sum of the double of the smaller integer and one; 2 the sum of both numbers’’ (Maillard 1960, p. 4).

In order to do so, it suggested completing a chart showing the square and difference of squares of integers 1 to 10, and analysing the succession of numbers thus obtained. Then, it suggested demonstrating successively the relations (a ? b)2 = a2 ? 2ab ? b2; (n ? 1)2 = n2 ? 2n ? 1; (n ? 1)2 - n2 = 2n ? 1, which would enable the pupil to justify his analysis answering the previous question. As we shall see in this example, here nothing tended to connect mathematical results with ‘‘real-life’’ situations. However, the approach that was used to give an entrance within a chapter which then adopted a more familiar form was quite remarkable: pupils were invited to observe and, if possible, have the intuition of a relationship that could be demonstrated afterwards. It was only in a second phase that the lesson 33

The name itself suggested an origin within physics, see above footnote 2.

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was organized as the successive statement of definitions, rules, and exemplified and carefully demonstrated proprieties and theorems. From one textbook to another, similar kinds of ‘‘practical works’’ indicate that this pedagogical method was not limited to calls for the use of ‘‘experience and practical reality’’ and the reinforcement of connections between mathematics and real life; it was also a means to introduce experimental thinking within mathematics, in which case mathematical knowledge was established through experiments and observation in a way quite similar to what happened in physics. As stated in the ‘‘thoughts on the mathematics syllabus for the troisie`me form’’ from the Cours R. Maillard, ‘‘practical works’’ were used ‘‘for introduction as well as for application’’ (Maillard 1960, p. v). Then, it was not only the extension of mathematics to applications within late 1950s mathematics textbooks which was at stake, but also the definition of methods and types of reasoning in mathematics. These could modify the identity of mathematics and its relationship with physics: through ‘‘practical works’’ a new legitimacy was granted to inductive reasoning and experimental approaches. However, such a shift in mathematics’ identity could appear contradictory with the deductive logical structure associated with the subject, and still embedded in pages’ layout and highlighted in authors and editors’ forewords (see Sect. 1). Finding out how textbooks authors solved this dilemma amounts to finding out whether for them ‘‘practical works’’ were merely a pedagogical method to introduce mathematical knowledge or whether they reflected a truly mathematical method, useful to develop new knowledge. For late 1950s France, answering this question leads to take into account competing conceptions of mathematics, directly related to the culture of the two orders that still permeated science teaching and school organization.

3 Reestablishing Differences: When the Audience Matters 3.1 The Case of Mathematics: Distinct Traditions for Distinct Audiences The introduction of ‘‘practical works’’ in mathematics syllabi in the late 1950s was aligned with the use of ‘‘active’’ methods advocated during the interwar years and more systematically experimented after the war in ‘‘Classes nouvelles’’ (Savoye 2010). In this regard, the innovation drew on previous experience. However, for some teachers and mathematicians whose ideas found a resonance within the French association of mathematics teachers of public secondary schools, the Association des professeurs de mathe´matiques de l’enseignement public [APMEP], it was linked to the promotion of the modernization of mathematics and introduction of ‘‘new math’’ contents at school (D’Enfert 2010). The APMEP gathered mainly secondary school mathematics teachers and its active members were in the 1950s influenced by the works of the Bourbaki group.34 They were convinced that all pupils should learn mathematics that were actually practiced within the university (see also Carsalade et al. 2013). For them, active methods were a means to put children in a situation in which they would have to work as mathematicians (which meant 34 Bourbaki is the name given to a group that young French mathematicians created in the 1930 s with the aim of reorganizing mathematics according to the central notion of structure. Bourbaki would have an important influence on the development of mathematics and (though rather indirectly) mathematics teaching after World War II.

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searching for solutions and exploring the mathematical world) and to introduce ‘‘new’’ math. Their conception of active methods was thus quite different from the one promoted since the interwar period which encouraged pupils to elaborate themselves solutions to problems and exercises under the teacher’s guidance. By introducing ‘‘practical works’’, late 1950s official directives for colle`ges and lyce´es reflected the consensus towards the use of ‘‘active’’ pedagogical methods, but they were also more particularly influenced by the APMEP’s conception. Indeed, mathematics teachers were invited to use ‘‘practical works’’ in a similar way as teachers who sought to introduce modern mathematics to their pupils. The idea that lay behind was that, thanks to a practical start, more children could gain access to the ‘‘abstract essence’’ of mathematics (D’Enfert 2010). In a context where the need to expand the recruitment of scientists, engineers and technicians was particularly stressed upon, the possibility of increasing the number of mathematics users and modernizing mathematical contents at the same time was full of promise. Despite the idea that the method also reflected ‘‘science in the doing’’, ‘‘practical works’’ were merely a means towards a mathematical world conceived as a formal, abstract and theoretical structure. As such, this understanding ran particularly counter to the mathematical culture of primary school mathematics teachers. The latter inherited indeed a conception of mathematics as a rather inductive and practical science, different from the abstract, theoretical and deductive conception that prevailed among secondary schools teachers (D’Enfert 2006; Assude and Gispert 2003). The line of demarcation between primary and secondary cultures of mathematics found its way within late 1950s textbooks, even though the latter were dominated by the secondary culture since most authors were lyce´es teachers agre´ge´s of mathematics. Besides, only one textbook series was exclusively dedicated to cours comple´mentaires pupils. It was written by the former pupils of the E´cole Normale Supe´rieure de Saint-Cloud Albert Lermusiaux and Marcel He´meret. In the forewords of the different volumes, they stressed the importance of ‘‘experience and practical reality’’ (He´meret and Lermusiaux 1958) and highlighted the interest of an experimental approach in order to meet the contradictory aims teachers were facing in cours comple´mentaires: ‘‘It must be acknowledged that traditional mathematics teaching was a bit too much intended for brilliant pupils, which is why it appeared as an uninterrupted succession of logical and rigorous deductions, with only a remote connection with everyday life. Yet most pupils of cours comple´mentaires will rapidly need useable mathematics. … However, teaching in cours comple´mentaires should not handicap pupils who would study further, especially future students at teacher schools. To try to reconcile these very different priorities, new mathematics teaching dedicates a very great space to practical works; these are useful to make mathematics teaching less austere, more practical while at the same time they give an opportunity to show children how useful mathematics is and how its applications have been developing’’ (He´meret and Lermusiaux 1960, foreword, authors’ emphasis).

For them, the role of ‘‘practical works’’ was threefold. Apart from their use as applications which connected mathematics with ‘‘real life’’ and their use as knowledge introduction which was considered in this case as a means to give mathematics teaching a ‘‘truly experimental character’’, ‘‘practical works’’ could also be entirely combined with the lesson (He´meret and Lermusiaux 1960, foreword). Within the books, such an idea was embedded in the similar graphical treatment of lessons and ‘‘practical works’’: ‘‘practical works’’ were not typeset in a specific chapter section, but they were rather directly inserted among the lessons. The different nature of the texts was only indicated by the use of a simple label on the top of the page signalling to the reader whether he was dealing with a

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lesson (the page was then labelled ‘‘lec¸on’’) or a practical work (the label was the short ‘‘T.P.’’ for travail pratique). Other textbooks authors and editors promoted quite a different conception of ‘‘practical works’’ and mathematics in their forewords. While they generally acknowledged the usefulness of the method, they linked it to pupils’ age and intellectual possibilities and praised its interest as a teaching tool in order to present a lively and attractive science. However, for them, it was merely a pedagogical means. Monge and Guincham (1958b), for instance, insisted on its usefulness at the beginning of mathematics study, but they stressed at the same time that an evolution was required as children grew up: ‘‘the sixie`me form has been a practical introduction to mathematics. From the cinquie`me onwards, the pupil begins more theoretical studies. It would nonetheless be premature to give mathematics … too abstract a form; some pupils might then turn away as it would appear a stern, difficult to understand, and even abstruse science. … Practical works present a feature different from the previous year; simple observations [constatations] are replaced by exercises of directed research, which present a greater intellectual benefit for older pupils’’.

For the authors of the Cours J. Marvillet, dangers of the method had to be pointed out to the teachers from the start: ‘‘In no instance, this attention towards the use of intuitive or experimental notions as starting points should make lose sight of the training of logical thought. … The purpose of practical exercises cannot be reduced to the intuitive discovery or to the verification of a property or a result. They shall aim at helping the pupil to progressively appreciate the necessity of rigour and the requirement of demonstration’’ (Girard et al. 1958, foreword).

Such a distinctive identity of mathematics also showed through the forewords written by the authors of the Cours R. Maillard when they noticed that using practical situations was a means to ‘‘rise’’ (s’e´lever) and reach mathematical precision and abstraction (Maillard 1958, p. 5, 1959, p. 7). Textbooks published for the next forms prompted this idea further, as they also made diverse positions regarding the teaching of ‘‘new math’’ apparent. While Paul Dubreil suggestively noticed that the book for the troisie`me form he had edited showed the authors’ refusal to ‘‘tone down’’ (e´dulcorer) mathematics (Dubreil 1963, p. 5), textbooks for the troisie`me forms of the Cours J. Marvillet were advertised in a more explicit manner. Among the main characteristics of the books, the publisher stressed in his catalogue the ‘‘exclusive use of deductive reasoning (except for solid geometry) and modern terminology’’, as well as ‘‘the prudent introduction of the notion of set’’.35 These indicated a commitment for the ‘‘new math’’ which was explicitly stated when the books were described as means to ‘‘prepare lyce´e pupils to a closer contact with the modern point of view which would only be studied in seconde form, and allow short streams pupils not to leave school while ignoring the new orientation of mathematics’’ (Librairie Armand Colin 1961). The Cours R. Maillard, which shared with the Cours J. Marvillet a conception of mathematics as an abstract, logical and deductive subject, insisted on the contrary of the novelty of the method rather than on the introduction of ‘‘new math’’ content: the entire 35 The Cours J. Marvillet was named after its editor and author Joseph Marvillet, agre´ge´ of Mathematics and teacher in the lyce´e of Strasbourg. Other authors contributed to this series, among which some were also agre´ge´s of mathematics (Lucette Chopard-Lallier or Robert Girard), but some were not: Pierre-Marie Fournier was head of a cours comple´mentaire and Alphonse Adam was inspector of primary instruction. The latter thus belonged to the primary order of instruction (and so was, to a certain extent, Robert Girard as he had been trained as a primary school teacher before passing the Agre´gation and joining the secondary order of instruction).

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series was advertised as a ‘‘proposition to renew the traditional manner both in the exposition of lessons and in the presentation of subjects’’ (Hachette 1960). All together, these assumptions positioned their authors vis-a`-vis ongoing debates on mathematics teaching. They also reflected a conception of mathematics more in line with secondary culture which was coherent with the fact that the series they wrote were primarily aimed at colle`ges and lyce´es pupils, while testifying to the growing influence of ‘‘new math’’ advocates even before new syllabi commanded the modernization of contents. However, these books also concerned cours comple´mentaires, either because from the quatrie`me form onwards the series included a version specifically dedicated to this type of school, or because they made no distinction between the different audiences.36 In the latter case, textbooks promoted the secondary culture towards the primary school audience as the only true one. Thus, late 1950s textbooks appear as tools who would encourage their users (both teachers and pupils) to approach mathematics in different ways. For the ones, experimental methods based on the manipulation and study of concrete objects and their physical properties would tend to be considered as a proper mathematical method. For the others, such means would be mere pedagogical methods, that may be necessary regarding the pupils age but which could not be typical of ‘‘true’’ mathematics. For the latter, the relatively important space devoted to real-world and physical problems within sixie`me and cinquie`me textbooks which blurred the boundaries with physics (see Sect. 2.1) needs to be understood as a step on a progressive path towards abstraction and theoretical problems, or even exactitude

3.2 From Mathematics to Physics: Changing Points of View Around Exactitude and Uncertainty As Monge and Guincham (1958a, Avertissement) pointed out from the start of their series, mathematics teaching had to be ‘‘sure and precise as it ought to be for an exact science’’. By contrast, with this idea of exactitude which permeated more generally mathematics’ textbooks, the physical world they pictured seemed to be flawed. This idea appeared particularly clear through the entire series named after Paul Dubreil: it was explicitly stated when geometry was taken up, all the more so because mathematical notions were introduced thanks to the description of physical objects according to the pedagogical method I presented above.37 In what can be considered as a ‘‘boundary work’’ between scientific disciplines (Gieryn 1983, 1999), the authors insisted on the necessarily imperfect representation of ‘‘geometrical beings’’ compared with their ideal nature, thus promoting a Platonic conception of mathematics: ‘‘The most skilled worker cannot make a perfectly right ruler. The best draughtsman cannot trace a perfectly straight line …. It is a natural tendency for human mind to forget such imperfections and to imagine beings which totally and absolutely possess geometrical qualities. These beings, which only exist as creations of our mind, are geometrical beings. …’’ (Brailly-Marchand and Fouche´ 1961, pp. 42–43).

36 Among the titles mentioned in this paper, the Monge & Guincham and Lebosse´ & He´mery series included distinct textbooks for cours comple´mentaires on the one hand and for colle`ges and lyce´es on the other hand from the quatrie`me form onwards. By contrast, the Cours R. Maillard and the Collection Paul Dubreil were addressed to all children. The Cours J. Marvillet stood in between as it fragmented the audiences in the troisie`me form only. 37

The idea was also explicitly stated in chapters dealing with measurements and measuring devices.

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Within this textbook, the uncertainty attached to physical objects and measurements even reached a very symbolic item. When presenting the metric system, the authors commented on a meaningful gap between theory and physical reality. They mentioned that even the standard metre stored at the Pavillon de Breteuil and carefully made of selected materials in order to limit any kind of distortion, which was at the root of the internationally adopted metric system, could not be subtracted from physical imperfection: ‘‘In reality the most precise measurements that had been recently made have shown that the standard metre from the Pavillon de Breteuil is shorter by 2 tenths of millimetre than the length resulting from this definition [of length unit]’’. (Brailly-Marchand and Fouche´ 1961, p. 53).

Physics textbooks offered a different treatment of uncertainty. They acknowledged uncertainty associated with measurements and quantification of physical phenomena, but instead of linking it with corrupted or imperfect representation of truly existing (even though only in human minds) ideas, they presented uncertainty as reality per se. As such, it required a careful investigation and a physical definition. Helping pupils to become aware of the physical meaning of uncertainty constituted an aim highlighted by textbooks authors. Together with the experimental method and inductive reasoning, this emphasis contributed to define physics’ identity. It was embedded in the central place devoted to measurement and scientific instruments and to detailed discussions regarding the proper use of the latter, according to the context and the experimentation’s purpose. It also was given first place in the textbooks which opened their physics section or devoted their introduction to the issue of ‘‘Uncertainty of measures in physics’’, as the series edited by Maxime Joyal or the one written by Andre´ Godier, C. Thomas and M. Moreau did, respectively, thanks to a 6-page long introduction and a 1-page long ‘‘notice regarding unit systems’’ (Delattre and Boue´ 1961a; Godier et al. 1960). In parallel with this physical sense attributed to ‘‘uncertainty’’, the meaning of ‘‘exactitude’’ changed from mathematics to physics textbooks. In the latter, exactitude was no longer a characteristic of mathematics but a physical notion. In its investigation, quantification was required, no longer in order to be able to reason in absolute terms as suggested by some mathematics textbooks (see Sect. 2.1.), but rather to make sense of results obtained by measuring devices according to the situation. As Legreneur and Peyraud (1960, p. v) pointed out: ‘‘When appropriate …, quantitative experiments allow measuring phenomena or, at least, finding an order of magnitude. Such experiments familiarize the pupils with the notion of measure; they show that a measurement result is never absolute, but always more or less approximate, which give rise to notions of uncertainty and measure exactitude’’.

Despite its important and acknowledged role, particularly fundamental in the construction of scientific notions presented as superior to sensation, quantification could not stand alone in the making of physical meaning. As Payan et al. (1960, p. 2) warned their readers: ‘‘Each experiment first has a qualitative aspect which suggests in which way the relationship between cause and effect works: it is then translated by a table of numbers which would be a mere report without any scientific interest if qualitative reasoning did not allow grasping the ‘‘correspondence’’ for which mathematical expression is generalization’’.

Compared with the hierarchy implicitly established between abstract mathematics and practical physics in secondary schools textbooks, here the situation is turned round. Mathematical formalization becomes a useful tool, but its significance remains limited: the physical meaning of phenomena remains profoundly dependent on more practical and qualitative approaches.

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The analysis of late 1950s physics and mathematics textbooks illustrates how, through such teaching tools, conceptions of distinct and separated disciplines might have actually been promoted. It also helps to understand why in educational contexts both subjects have gained identities which tended to reduce their mutual interplay to reciprocal usefulness. However, within these past textbooks, the interplay of mathematics and physics is not limited to reciprocal usefulness: some subjects and pedagogical methods actually put into question the sharpness of disciplines’ boundaries. While for some authors of mathematics textbooks, the blurring of boundaries resulted from a pedagogical necessity and for others it came from the possibility to use the same kind of reasoning in both disciplines. For these different authors, the hierarchy between disciplines and the sense given to notions as exactitude, uncertainty or quantification were not the same. In order to understand this variety, it is essential to take into account the two-tier organization of French schooling of the time. As given in Sect. 3.1., such an organization gave way to different and, at the time, even competing conceptions of mathematics. While physics textbooks could be seen in opposition with mathematics textbooks promoting the secondary culture of mathematics, they were no longer so much opposed to mathematics textbooks written in the primary tradition. Yet, the audience aimed at by physics textbooks was precisely constituted of pupils and teachers belonging to this primary tradition: only pupils accommodated in cours comple´mentaires and in short modern streams of colle`ges and lyce´es studied physics in middle schools. These pupils were prepared to leave school after the 4 years of post-elementary studies either to start working or to be trained in vocational, often technical, schools. Accordingly, their projected needs were an important parameter which shaped their education, but it resulted in a peculiar practical and technical conception of science, which concerned physics and mathematics. These two subjects could not be thus considered, in this particular tier, as strongly independent, quite the contrary. Subsequent reforms of the French national education system and, in particular, the Haby reform which promoted a unique secondary type of schools to cater for pupils aged between 11 and 15 have contributed to the disappearance of this original tradition. Taking it back to light shows first that alternative conceptions of mathematics and physics identities and, accordingly of the interplay of both disciplines, could have actually existed in a single country at a same moment. It reminds of approaches that might have been forgotten and which still could have some value nowadays, in order to understand the difficulty to promote a single and culturally loaded conception of disciplines as the only true one, and in order to think of innovative ways of teaching. Indeed, the different conceptions found in the late 1950s textbooks and the contemporary debates highlight the relationships between conceptions of the audience aimed at by textbooks and teaching one the one hand and conceptions of scientific disciplines on the other hand. Acknowledgments I wish to thank Laure Pellet and Laura Dang for the information they gave me regarding primary instruction pupils and their teachers, Ricardo Karam for his comments on an earlier version of this paper, and the anonymous referees whose remarks and suggestions helped to improve it. I am also grateful to Michael Matthews for his patience in waiting for this paper. I also wish to thank the publishing houses which kindly permitted the reproduction of the images illustrating the paper.

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