Mathematics teaching in French écoles normales primaires, 1830–1848: social and cultural challenges to the training of primary school teachers

In the nineteenth century, the French education system, and teaching itself, was organized according to the social class the pupils came from: primary education was for the working classes, and secondary education was for the wealthier classes. In relation to this ‘educational duality’ this paper looks at the mathematics teaching provided in the écoles normales primaires (primary teacher training colleges), which developed in France in the 1830s to train future (male) primary school teachers. What, precisely, was the content of this teaching? How was it organized? In what spirit and for what purpose was it provided? The aim of the paper is to show how these schools participated in the construction of a specifically primary mathematics culture for the education of the children from the lower classes, as distinct from the scholarly culture of secondary education.


Introduction
In nineteenth century France, the teaching of mathematics in schools was structured-with regard to content and its methods as well as its aims-by the organization of the school system into two separate, parallel networks. On one hand, primary education catered for the working classes: it offered a practical education, a preparation for the daily or professional life for which its pupils were destined. On the other hand, secondary education aimed at children of the wealthy, that is to say a small minority, 1 who were sent to the royal and communal colle`ges: in these colle`ges (which included elementary classes), pupils were given a humanist education, whose purpose was the intellectual, 'disinterested' development of the mind. Unlike today, there was no integrated educational system in which primary schools would make up the first stage and secondary schools the following one: education replicated social stratification by separating primary schooling from secondary schooling and children of the lower classes from those of the bourgeoisie.
Until now, most historical research has focused on mathematics teaching in secondary education, and one has to highlight with regard to this subject the importance of Bruno Belhoste and Hélène Gispert's research works (Belhoste 1989(Belhoste , 1995Gispert 2002;Gispert et al. 2007). In contrast, there are far fewer studies on mathematics teaching in French primary schools, and in particular in what we could call 'higher' primary education, that is to say e´coles primaires supe´rieures (higher primary schools) and e´coles normales primaires (primary teacher training colleges) which formed, at the time, an extended and parallel education to that of the secondary system.
In an article written in 2006, Gert Schubring underlines the necessity of undertaking research into the period when the teaching of mathematics was institutionalized, with particular interest given to the teachers and their training (Schubring 2006, p. 675). It is from this perspective this paper describes and analyzes the mathematics teaching R. d 'Enfert (&) Ecole normale supérieure de Lyon, UMR 5190 LARHRA (équipe Histoire de l'éducation), Lyon, France e-mail: renaud.denfert@freesbee.fr given in e´coles normales primaires created in France in the late 1820s and early 1830s to train young men to be lower and higher primary school teachers. What precisely was the content of this teaching? How was it organized? In what spirit and for what purpose was it provided? At a time when the State was extending its control over primary education-establishing it as a 'public service'and when its efforts focused on male education, these new schools played a key role in creating a specifically primary mathematics culture, distinct from that of secondary education.
2 The first network of e´coles normales primaires and its institutional environment During the first half of the nineteenth century e´coles normales primaires were created in order to train (male) primary school teachers (Grandière 2006;Condette 2007). The guiding principles were established under the reign of Napoléon I by the 17 March 1808 Imperial Decree. This decree created the University, which was both the administration of the new educational system and the body of teachers in secondary schools and higher education. Primary education and primary school teachers were a marginal part of the University. Nevertheless, the 1808 decree provided that there would be ''created in every academy […] one or several 'classes normales', aimed at training primary school teachers. The best methods to improve the teaching of reading, writing and arithmetic will be introduced.'' 2 A classe normale opened in Strasbourg in 1810 and soon became an autonomous school. 3 Under the Bourbon regime (1815-1830), a constitutional monarchy, other cities in the east of France followed the Strasbourg example in the early 1820s, but it was not until the end of the decade that a network of e´coles normales primaires really took shape. A dozen e´coles normales were created in 1828-1829 under the decisive impetus of the Minister of Public Instruction Antoine de Vatimesnil. This creative movement gained momentum following the July 1830 revolution and King Louis-Philippe's accession to the throne, notably with the appointment of François Guizot as Minister of Public Instruction. Under the 28 June 1833 Guizot Law on (male) primary education, each French de´partement had to maintain an e´cole normale primaire. In 1833 there were 47 such schools and by 1840 there were 76, training 2,684 student-teachers, which gave an average of 35 student-teachers per school. The student-teachers generally came from rural backgrounds: sons of farmers, rural artisans, smallholders and countryside teachers. Although women's education is not the subject-matter of this paper, it should be noted that that e´coles normales for young women were not given the same level of attention by the government of the day. Their creation was not made compulsory by the legislator (this only happened in 1879), which means they formed a far smaller network in comparison with the e´coles normales for young men: there were only eight e´coles normales for young women in 1848, to which we have to add some thirty classes normales, annexes of higher primary schools or boarding schools. The increase in the number of e´coles normales primaires took place in a new political context, following the victory of the liberals over the conservatives which resulted in a liberal constitutional monarchy in France under King Louis-Philippe after the July 1830 revolution (the beginning of the 'July Monarchy'). Increasing the number of e´coles normales was part of a broader development policy, with the State taking over and running primary education, a quarter of a century after it had taken over secondary education. For the new government, education of the people in primary schools was to guarantee the stability of the state and society. In a letter to primary school teachers, in which he also reminded them of their modest social status, Guizot wrote: Although the career of a primary school teacher is dull, although his days are spent in the commune where he also does his teaching, his work is of interest to the whole society, and his profession plays a significant role in civil service. It is not only for the commune, nor in purely local interest that the law requires that all French citizens acquire, if possible, the knowledge essential for a social life, and without such, intelligence languishes and sometimes stupefies: it is also for the State and in the public interest; because freedom is only ensured and constant for people enlightened enough to listen to the voice of reason in all circumstances. Universal primary education from now on is a guarantee of social order and stability. As everything in our Government's principles is true and reasonable, to develop intelligence and spread enlightenment, is to protect the influence 2 ''il sera établi auprès de chaque académie […] une ou plusieurs classes normales, destinées à former des maîtres pour les écoles primaires. On y exposera les méthodes les plus propres à perfectionner l'art de montrer à lire, à écrire et à chiffrer.'' Most of the official texts concerning mathematics teaching in primary school between 1791 and 1914 have been published in d' Enfert (2003a); those relating to the e´coles normales primaires in the same period in Grandière et al. (2007). 3 Classes normales and then e´coles normales primaires must not be mistaken for Ecole normale supe´rieure which was also created in 1808 for the training of teachers for secondary schools and opened in 1810. and the continuity of the constitutional monarchy (Guizot 1833, p. 269). 4 From 1831, a series of five official textbooks were published by the Ministry of Public Instruction and sent to primary schools. Amongst them was Petite arithme´tique raisonne´e by Hippolyte Vernier, of which 25,000 copies were distributed in 1832, and 30,000 in 1833 (Choppin 1993, p. 31-32). 5 In 1832, Guizot, then Minister of Public Instruction, decided to create a monthly review, the Manuel ge´ne´ral de l'instruction publique, aimed at guiding primary school teachers in their work. Above all, the 28 June 1833 Guizot Law made it compulsory for every commune to maintain at least one primary school (for boys), in which would be taught ''moral and religious education, reading, writing, elements of the French language and arithmetic, the legal system of weights and measures.'' 6 The law also provided for the opening of higher primary schools in the main administrative centers and towns with over 6,000 inhabitants, to offer the children of the middle classes a more developed education than the one given in lower primary schools, more orientated towards science and its practical applications in comparison with secondary schools. In addition to the subjects taught in lower primary schools, higher primary education had to include: ''elements of geometry and their usual applications, especially linear drawing and land surveying, some notions of physical science and natural history that are applicable to everyday life, singing, elements of history and geography, and above all the history and geography of France.'' 7 In 1835 an inspectorate of regional primary school inspectors was created, supported by sub-inspectors from 1837, with the aim of monitoring pedagogical standards in primary schools.
As part of the primary education system created by the Guizot Law, the e´coles normales primaires were subject to a whole set of regulations concerning their operation, organization, and the content of the training provided, as well as the certification of the school teachers. Studentteachers, who had to be over 16 years old, were recruited through a competitive examination, where they had to prove, in particular, that they possessed ''basic notions of arithmetic''. In theory their training lasted 2 years, but from the end of the 1830s a number of schools offered three-year courses. Far from being reduced by the 1808 decree to 'reading, writing and arithmetic', the teaching they received was composed of numerous subjects which, with a few exceptions, followed the requirements of the Guizot Law very closely. Pedagogical training (teaching methods and principles of education) was also provided. The student-teachers had to pass a brevet de capacite( teaching ability certificate) examination before they were allowed to work as primary school teachers (Toussaint 2002). There were two distinct types of certificate, corresponding to the two primary school levels established under the Guizot Law: the elementary teaching certificate for lower primary schools and the higher teaching certificate for higher primary schools. However, the latter was only obtained by a small minority of the student-teachers (130 out of 860 in 1840, which was only 15 %) (Villemain 1841, p. 110). Graduated teachers in lower (respectively higher) primary schools were then supposed to teach all subjects sanctioned by the elementary (respectively higher) teaching certificate they held.
The first e´coles normales primaires did not have a body of trained teacher-trainers at their disposal. Consequently the teacher-trainers came from diverse backgrounds (d 'Enfert 2012). In general they either came from the world of primary education, in which case they were primary or higher primary school teachers or headmasters, or even school inspectors or sub-inspectors, before being employed in the e´coles normales; or they came from the world of secondary education, that is to say they were (and in many cases continued to be) teachers in royal or communal colle`ges. However, some teacher-trainers at the e´coles normales primaires came neither from primary nor secondary education: in this instance, when speaking of mathematics, the teacher-trainers were often architects, draftsmen, or even land surveyors. In fact, for many the teacher-training was combined with a statutory job in another school in the town (higher primary or secondary school, or art school), or in a local administration, that offered them an additional income. In Chartres, around 1845, the colle`ge mathematics teacher gave ten 2-h classes 4 ''Bien que la carrière de l'instituteur primaire soit sans éclat, bien que ses soins et ses jours doivent le plus souvent se consumer dans l'enceinte d'une commune, ses travaux intéressent la société tout entière, et sa profession participe de l'importance des fonctions publiques. Ce n'est pas pour la commune seulement et dans un intérêt purement local, que la loi veut que tous les Français acquièrent, s'il est possible, les connaissances indispensables à la vie sociale, et sans lesquelles l'intelligence languit, et quelquefois s'abrutit : c'est aussi pour l'É tat lui-même et dans l'intérêt public ; c'est parce que la liberté n'est assurée et régulière que chez un peuple assez éclairé pour écouter en toute circonstance la voix de la raison. L'instruction primaire universelle est désormais une des garanties de l'ordre et de la stabilité sociale. Comme tout, dans les principes de notre gouvernement, est vrai et raisonnable, développer l'intelligence, propager les lumières, c'est assurer l'empire et la durée de la monarchie constitutionnelle.'' 5 However these figures are to be compared to the figures concerning another official textbook, L'Alphabet et premier livre de lecture, Paris, Hachette and Firmin Didot, 1831, with 500,000 copies distributed in 1831, 200,000 in 1832 and 300,000 in 1833. Those books were to be freely distributed to the poorer pupils. 6 ''l'instruction morale et religieuse, la lecture, l'écriture, les éléments de la langue française et du calcul, le système légal des poids et mesures.'' 7 ''les éléments de la géométrie et ses applications usuelles, spécialement le dessin linéaire et l'arpentage, des notions des sciences physiques et de l'histoire naturelle applicables aux usages de la vie, le chant, les éléments de l'histoire et de la géographie, et surtout de l'histoire et de la géographie de la France.'' per week in the e´cole normale (for 1,400 francs per month), in addition to the eight classes he was required to give in the colle`ge (for 1,800 francs per month). This raises the question for those who were also colle`ge teachers of whether their lessons complied with the spirit of 'primary' education, or if, on the contrary, they remained faithful to the standards of secondary education. To reduce the heterogeneity of the teachers, the Ministry of Public Instruction tried to limit the recruitment of colle`ge teachers to those from primary education, notably by favoring the former students from the e´coles normales. However, this does not seem to have had the desired effect as until the mid-1840s there was still a large proportion of teachertrainers from secondary education (nearly 40 % in the case of mathematics).
Around 1840, students from the e´coles normales filled about one third of the vacant teaching positions (Grandière 2006, p. 62). But one has to remember that the e´coles normales did not only provide the initial training for primary school teachers. A good number also provided in-service training for teachers already in duty, by organizing special pedagogical conferences and refresher courses, for example on the metric system which use became mandatory in 1840. Through expanding the teachers' knowledge further than only 'reading, writing and arithmetic' the e´coles normales primaires played a key role in improving primary education and in building a true professional teaching culture. By doing so, they contributed to shaping the mathematics culture and its representations of a great many individuals of the nineteenth century French society.
3 Which 'mathematics' for the training of primary school teachers?
Contrary to secondary teaching, 'mathematics' was not a specific discipline in primary teaching, even if the word was sometimes used by those concerned. One can, nevertheless, define the content of what was taught from the official texts published at the beginning of the 1830s (d 'Enfert 2003a). According to the 14 December 1832 rules and regulations for e´coles normales primaires, they included ''arithmetic, including the legal system of weights and measures'', 8 as well as ''linear drawing, land surveying and other practical applications of geometry''. 9 But the Guizot Law later defined, as we have seen, ''elements of geometry and their usual applications, especially linear drawing and land surveying'' among the disciplines of the higher primary schools which had to be created. As a result, geometry (and not only practical geometry) became a fully fledged subject for primary education, and therefore of teacher training. In fact, geometry was introduced into the primary school mathematics curriculum after the debate on the Guizot Law in the Chamber of Deputies (29 April 1833). The main point was that linear drawing 10 -whose teaching had been introduced progressively into primary schools since 1818-was based on the drawing of geometric lines and figures. It thus followed the Louis-Benjamin Francoeur method (1819) for the e´coles mutuelles using the monitorial system. Several Representatives from scholarly backgrounds asserted that linear drawing was no more than an application of geometry in the same way as land surveying was: it thus became necessary, in their opinion, to teach the theoretical principles on which these applications were based, that is to say the ''elements'' of geometry. Under the Guizot Law, geometry thus provided the theoretical base for linear drawing as well as land surveying (d 'Enfert 2003b).
In total, arithmetic and the metric system, geometry, linear drawing and land surveying (to which ''toise´''measurement-and producing survey plans, also considered as ''usual applications'' of geometry, have to be added) made up the four major components of mathematics teaching in the e´coles normales primaires. 11 However, in the majority of these schools, mathematics was not taught by one teacher, but by several teachers: often two, sometimes three, and exceptionally four (d 'Enfert 2012). In the e´cole normale in Guéret around 1845, two teacher-trainers shared mathematics teaching: the first was responsible for arithmetic, the second for geometry and linear drawing. In the e´cole normale in Chartres, one teacher gave lessons in arithmetic and geometry, another taught linear drawing, and a third land surveying. In certain e´coles normales, linear drawing (possibly leading up to land surveying and producing survey plans) was taught by a technical drawing specialist, such as an architect or a draftsman for instance, which, contrary to the legislators' intentions, means it was taught 'separately', detached from the main geometry lessons as such. The sharing of mathematics teaching between several teacher-trainers has a corollary: numerous teachers did not only teach mathematics but also gave lessons in other disciplines such as physics, natural history, mechanics, or even reading, writing (or calligraphy), grammar, history or geography. While in secondary education mathematics constituted a priori a unique and relatively autonomous disciplinary entity, the fact that it was here taught by several (and possibly versatile) teachers gave a fragmented image of the discipline to the future primary school teachers.

Mathematics teaching focusing on applications useful in daily life
Primary school was the school of the lower and working classes: its teaching had therefore to be relevant to the social and professional paths of its pupils and offer a practical, concrete, ordinary education which answered the needs of their daily life and future professional activity (d 'Enfert 2003a'Enfert , 2007. In this context, ''the aim of the e´coles normales primaires is to train school teachers, above all village school teachers: their knowledge must be solid, practical, and likely to be transmitted in the form of teaching that is immediately useful to the men whose laborious condition deprives them of the necessary leisure time to reflect and study'' (Guizot 1834, p. 87). 12 The tension was then strong between, on one hand, the ''willingness to give the teachers an identity linked to their knowledge'' (Grandière 2006, p. 55) 13 and, on the other hand, the fear that these teachers, now educated, would aspire to higher ambitions which had no relation to the modest position to which they were a priori destined. One thing is certain: the teaching in e´coles normales was to be distinct, both in the form and spirit, from secondary education.
Mathematics teaching was directly concerned in this tension. The ministerial administration permanently monitored the discipline to make sure it did not approach the more theoretical, abstract and also more speculative secondary school pedagogical model-and therefore made sure it kept its utilitarian and practical function. Mathematics teaching had therefore to be limited, according to the minister of Public Instruction Abel-François Villemain, ''to the most essential elements, amongst those which are the most immediately applicable to daily life'' 14 and ''neither the details on logarithms, nor the lessons in algebra, nor programs drawn from Legendre's Geometry'' 15 were allowed (Villemain 1843, p. 211). In practice, some e´coles normales went much further than the 'primary' framework defined in the official texts. In the e´cole normale in Nancy for example, trigonometry and algebra (up to quadratic equations) were taught, which was considered ''too much for future primary teachers'' (General Inspection 1838). Geometry, an eminently speculative discipline-which, as we have seen, was freshly introduced into primary education-was held under particular surveillance: it had to be ''reduced to its simplest expression'' (Villemain 1843, p. 211). And indeed, what was important was less the sequence of the propositions and the rigor of the proofs than the statement of the most 'useful' theorems. To keep geometrical teaching within the 'correct limits', the General Inspectors (of secondary education) who visited the e´coles normales primaires each year promoted school textbooks more specifically written for primary education-the production of which books expanded greatly during the 1830s and 1840s. Similarly the General Inspectors urged the teacher-trainers to give up Legendre's Geometry textbook. It was still tolerated in the libraries of the e´coles normales primaires but was considered too emblematic of secondary education mathematics which favored rigorous, hypothetic-deductive reasoning over the more practical and elementary textbooks such as Bergery's (1831), Desnanot's (1835) and Vernier's (1830). Thus in the Journal de l'instruction e´le´mentaire, a publication founded by people close to Guizot, we can read about the Vernier textbook: ''M. Vernier, in his Geometry textbook, which has only 179 pages, defines the most useful theorems, the most interesting problems […]. There is less rigor in the proofs than in Legendre's or Vincent's Geometry textbooks; but the students understand it much more easily: thus was the aim of the author, an aim which he reached'' (Anonymous 1831). 16 Bergery's Ge´ome´trie des e´coles primaires is emblematic of how some textbooks gave place to practical applications relating to construction and craft industry, as this problem witnesses: Place a piece of timber according to a given slope. Draw, on the construction site, two perpendicular lines ac, 12 ''le but des écoles normales est de former des maîtres d'école, et surtout des maîtres d'école de village : toutes leurs connaissances doivent être solides, pratiques, susceptibles de se transmettre sous la forme d'un enseignement immédiatement utile aux hommes que leur laborieuse condition prive du loisir nécessaire pour la réflexion et l'étude.'' 13 ''la volonté de donner aux instituteurs une identité liée à leurs savoirs.'' 14 ''aux éléments les plus essentiels, parmi ceux qui sont le plus immédiatement applicables aux usages de la vie.'' 15 ''ni les détails sur les logarithmes, ni les leçons d'algèbre, ni le programme tiré de la géométrie de Legendre.'' 16 ''M. Vernier, dans sa Géométrie, qui n'a que 179 pages, donne les théorèmes les plus utiles, les problèmes les plus intéressants […]. Il y a moins de rigueur dans les démonstrations que dans la Géométrie de Legendre ou dans celle de Vincent; mais les élèves comprennent avec bien plus de facilité : tel est le but que s'est proposé l'auteur, et qu'il a atteint.'' Contrary to those by Bergery and Desnanot, the Vernier textbook was also destined for arts and humanities classes in secondary schools.
bc. If the roof slope is 68 centimeters per meter, as flat tiled roof rafters might be in this area, mark the point c to a meter from b and 68 centimeters from a; then draw a line ab which is the direction of the piece of timber. For a very precise direction, one might instead of one meter, mark the point c to 3 or 4 meters from b and 3 or 4 times 68 centimeters from a: the slope would be the same, points a, b would be farther apart, and the small mistake made while placing the rule or the string at the points would change much less the prospected direction. (Bergery 1837, p. 58). 17 How can we explain that the teacher-trainers in some e´coles normales primaires went beyond the limits imposed by the official texts? While some reproduced the standards of secondary education on their own initiative, others answered to local demands. In the e´cole normale in Charleville for example, the surveillance committee asked the geometry teacher-a primary school teacher-to teach from a mathematics textbook for the candidates to the Ecole Polytechnique, in which the knowledge of algebra was a prerequisite to lessons in trigonometry (General Inspection 1838). Similarly, an 1843 report pointed out the very high demands made by certain examination commissions for the brevet de capacite´, due to the presence in the commissions of engineers and graduates from the Ecole Polytechnique. The latter would give way to ''the vanity of showing off their own knowledge rather than revealing the candidates' one'' (Beudant 1843), 18 and thus would urge teacher-trainers to go beyond the limits of the program so that their students could answer the questions asked by the commissions.
In fact, it is essentially in the relationship between theory and practical applications that the difference between the mathematics teaching in secondary schools and that of the e´coles normales can be understood: in the first, theory dominated practice; in the second, the practice and its applications had to take precedence. So, when teaching in e´coles normales primaires, secondary school teachers would have to adapt their teaching practice to the primary education standards. In Tulle, for example, mathematics teaching was ''too close to the habits developed in the collège. It is more theoretical than practical'' (General Inspection 1849). 19 In this respect, we must underline the essential role played by linear drawing, land surveying, producing survey plans, etc., in geometry teaching in the e´coles normales. This graphic teaching allowed diverse notions of geometry written in the program to be dealt with intuitively and concretely, and, in doing so, ''enlighten theory through practice''. On one hand, graphic activities were used to prepare for the geometry-only lessons: in the e´cole normale in Limoges, for example, the linear drawing classes helped the first-year students to learn the principal definitions of geometry as well as the drawing of geometric figures from the simplest to the more complex oval, ellipses, volutes, etc. (General Inspection 1838); in the same way, in Strasbourg, a large part of the first-year class was devoted to ''the theoretical presentation of linear drawing, the methods of constructing the different lines and figures, and the understanding of the properties of these figures. They must be used for the introduction of the study of geometry in the following year'' (General Inspection 1843). 20 On the other hand, graphic activities served as exercises for the application of the theoretical notions studied. Thus, some geometry textbooks intended for the e´coles normales primaires-and this was their essential difference from the classic books-offered graphical applications such as the drawing of moldings, paving/tiling and parquetry (Sonnet 1845) (Figs. 1, 2).
Exercises in producing survey plans, which combined drawing and practical geometry, were the crowning achievement of the geometry class and were taught at the end of the course. In the e´cole normale in Amiens for example, second-year student-teachers had to ''put into practice, in the field, lessons they were given on land surveying, leveling and geodesy. They produced the survey plan with the help of a graphometer, a compass and a plane table'' (General Inspection 1838). 21 And besides, such practices received strong high-level support. In Corsica for example, local authorities offered an award to ''the student who will be the best in land surveying, leveling and road layouts'' (Rendu 1838, p. 62). 22 A similar logic encouraged the Minister of Public Instruction to organize a linear drawing competition in 1838, in which the most advanced final-year student-teachers had to produce a survey plan of the buildings and lands of their e´cole normale (Fig. 3).
5 Substantial teaching in mathematics, but limiting geometry to a small elite The first regulations concerning the e´coles normales primaires, as the Guizot Law itself, simply set out a list of the subjects to be taught: the program of what had to be taught was not detailed. It therefore fell to local bodies, notably the surveillance committee in every e´cole normale, to establish the program and the timetable for the classes, as well as their distribution over the two (or three) year course. However, the will of the Ministry of Public Instruction to standardize the teaching for all the e´coles normales primaires urged them to define the content and timetable for these classes more precisely. In 1835 the Ministry of Public Instruction published a list of reference books for inclusion in the libraries of the e´coles normales, and a detailed national program for certain subjects the following year. Arithmetic and geometry programs were published in 1838. 23 They planned eighty 2-h arithmetic lessons (from numeration to the elementary notions of logarithms) and sixty 2-h geometry lessons (from plane and spatial to the measurement of surface areas and volumes) over a two-year course, to which was added a 1-h study time for each lesson. 24 However, no detailed program or timetable was published for the ''usual applications'' of geometry, such as linear drawing or land surveying. In fact, the two programs for arithmetic and geometry published in 1838 reproduced for a large part, in their content and wording, the programs of the 4th, 3rd and 2nd grades (14-16 years old) 25 of the royal colle`ges that had Fig. 1 Hippolyte Sonnet, Premiers e´le´ments de ge´ome´trie avec les principales applications au dessin line´aire, au lever des plans, a`l'arpentage, etc., 1859 (6th ed.): pl. 8-9 23 Programs were also published for the singing examination (29 March 1836), for physics, chemistry and mechanics (18 July 1837) and for history and geography (11 September 1828). 24 Some e´coles normales primaires also gave accountancy classes. 25 The student-teachers of the e´coles normales primaires were older than the pupils in secondary education who followed these programs.
been defined a few days before (Belhoste 1995, pp. 148-152). But the resemblance stops there. Not only were the programs different in their spirit, as we have seen above, but they also differed in the time they allocated to mathematics teaching: the program for the e´coles normales was made up of 140 lessons over the 2 years at the rate of two lessons per week (that is to say 4 h per week), whereas the far denser program in the royal colle`ges was made up of 60 two-hour lessons (40 for arithmetic, 20 for geometry) over 3 years at the rate of one lesson per week. If one adds the time assigned to the teaching of ''usual applications'' of geometry-generally 4 h per week according to the timetables kept in the archives-as well as the time assigned to study and practice, one can estimate that, for the student-teachers, 10 h were allotted to mathematics weekly, which is far more than in secondary schools. Thus appears another constitutive characteristic of the duality between primary and secondary schools which prevailed during the nineteenth century. Whereas secondary teaching, which privileged the classic humanities, placed mathematics (and more broadly speaking science) on the margins, the e´coles normales (and more generally 'higher' primary education) on the contrary offered the possibility, as long as the course was followed through to the end, of a large amount of practice in the mathematics studied from the point of view of its applications. So much so that, if one believes certain general inspectors, some e´coles normales became very attractive in regard to their science teaching: ''It appears that certain students [of the e´cole normale in Vesoul] have little interest in the profession of teaching, and that the particular care given to the study of science would above all come from the desire to enter into other services'' (General Inspection 1847). 26 The place held by geometry in the course constitutes another difference between the e´coles normales primaires programs and the royal colle`ges programs. In the e´coles normales (and contrary to comparable classes in the royal colle`ges), the arithmetic class, taught during the first year, had to be completed before starting the geometry classes. Those occupied most of the second year and possibly the third year of study, the student-teachers being then simultaneously ''trained to carry out the usual applications of arithmetic, as long as the lessons subordinate to geometry, land surveying, measurement and the elements of science will offer them the opportunity to do so'' (Conseil royal 1838, p. 483). 27 Of course, it was relatively common during the nineteenth century for arithmetic teaching to precede geometry teaching. But in the case of the e´coles normales, the organization of mathematics teaching was greatly dominated by the preparation for the brevet de capacite´: a degree whose exams and their content were strongly interrelated with the mathematics taught in lower and higher primary schools.
The first year of the e´cole normale was dedicated to the brevet e´le´mentaire (elementary teaching certificate) preparation, whose mathematics examination focused only on ''the elements of calculation'' and the metric system (and also linear drawing from 1841); the second year was a preparation for the brevet supe´rieur (higher teaching certificate) that demanded, amongst others, notions of geometry (angles, perpendiculars, parallels, surface areas of triangles, polygons, circles, volumes of the simplest elements) and the complementary arithmetic knowledge (proportions, the Rule of Three and the Rule of Fellowship), as well as linear drawing, land surveying, measurement, and producing survey plans. A Ministry of Public Instruction circular published on 9 August 1838 even restricted the second-year lessons exclusively to the students capable of obtaining the brevet supe´rieur, which excluded a large majority of student-teachers from the geometry lessons scheduled in the 1838 program. Thus, the headmaster of the e´cole normale in Châlons proposed that only the twelve best second-year students be allowed to study the subjects of the higher level (geometry, physics), either because they were deemed ''qualified to aspire to the brevet supérieur'', 28 or because they had been ''recognized capable of benefiting from these lessons without detriment to their elementary education''. 29 He planned, however, that ''elementary and practical lessons in geometry applied to land surveying'' would be given to all second-year students, ''as it is understood that no student can be deprived of this kind of knowledge, which they have sometimes the opportunity to use in order to add to the little resources owned from their position as a teacher'' (Guerrier de Haupt 1846). 30 While we can see here an allusion to the teachers' modest salary and to the fact that land surveying represented extra revenue for them, what appears above all, within primary education itself-and here within the e´coles normales-is that geometry had to be exclusive to a small elite of good students. Restricting the access to geometry teaching to a few means limiting, for the vast majority, mathematics teaching to its 'elementary' aspects (arithmetic, the metric system and, possibly, practical geometry). This had the effect of preventing the dangers of too broad an education which would change the vast majority of the student-teachers into ''semi-scientists'' far less inclined to live the ''vulgar existence'' of the countryside teachers. Despite it being part of the official program, geometry did not appear to be legitimate knowledge in the mathematics training of the teachers due to the social dangers it represented. However, we must point out that this kind of organization for mathematics teaching, which tended to separate arithmetic and geometry completely and to restrict the latter to the best student-teachers, was not always respected. In the e´cole normale primaire in Bar-le-Duc for instance, the study of geometry was mandatory for all students, and that also included first-year students. Its headmaster reported: ''there is no other better discipline to train the young intelligence'' (General Inspection 1843). Furthermore, the general organization for mathematics teaching began to arouse criticisms by the 1840s. In an 1843 report to the minister, a general inspector concluded that this organization ''destroys the spirit of the institution'' and advocated ''a normal two-year course for all students, regardless of the certificate they would later obtain'' (Beudant 1843). 31 In 1847, another report, emanating from a ministerial commission in charge of the programs' updating, questioned ''the fact of teaching classes in the first year that are given up in the second or third year […] especially arithmetic which is one of those classes for which it is important to continue the teaching over the whole period of study.'' (Rendu 1847, p. 238). 32 The commission, who wanted to extend the program on a threeyear basis, suggested that the teaching of arithmetic and geometry be done side-by-side over the first 2 years (at the rate of two weekly lessons of 1 h and a half), with the third year being devoted to the applications of geometry-land surveying, producing survey plans, measurement of surface areas and volumes-at the rate of two lessons per week. 33 However, those propositions could not be applied immediately, due to the political upheaval that happened at the end of the 1840s.

Conclusion
The July Monarchy (1830-1848) was a key period in the constitution and structuring of primary mathematics teaching. The e´coles normales primaires played a key role in this process: it was largely through the structure of the teacher-training that the required content of this teaching was made explicit, along with the approach to be taken and the principal targets to be achieved. Thus a specific school mathematics culture was outlined, largely determined by the social ends of primary school education, and strongly differentiated from secondary education. This culture would embrace new sciences such as geometry and its applications, which nonetheless had to remain practical, tangible and user-friendly, so as to answer the needs of daily and professional life.
The conservative reaction that followed the 1848 revolution would change this order. From then on there was mistrust of teachers and the e´coles normales, fuelled by the criticism which had become apparent since the early 1840s. Considered as 'pockets of socialism', these schools were even threatened with abolition. They were blamed for, amongst other things, extending their program too much, thus training young people to become too learned, less inclined to accept the modest (social) conditions of countryside school teachers. ''I prefer the church-bell ringing school teacher to the mathematician school teacher'' declared the Member of Parliament Adolphe Thiers (Chenesseau 1937, p. 32). 34 In the end, e´coles normales continued but with new regulations under the 15 March 29 ''reconnus capables de profiter de ces cours sans détriment pour l'instruction élémentaire.'' 30 ''attendu qu'aucun élève ne peut être privé de ce genre de connaissance, dont ils ont quelquefois l'occasion de se servir pour ajouter aux ressources trop minimes de leur position d'instituteur.'' 31 ''un cours normal de deux ans pour tous les élèves sans se préoccuper du brevet que chacun d'eux peut ensuite obtenir.'' 32 ''l'usage de faire dans la première année des cours dont il n'est plus question dans la seconde ni dans la troisième […] notamment pour l'arithmétique, quoique ce cours soit un de ceux pour lesquels il importe le plus que l'enseignement se prolonge pendant toute la durée des études.'' 33 The commission also suggested giving linear drawing lessons three times a week during the first and second year and two lessons a week in the third year. 34 ''J'aime mieux l'instituteur sonneur de cloches que l'instituteur mathématicien.'' 1850 Falloux Law, named after the Minister of Public Instruction at the time. Mathematics training of the future school teachers was profoundly modified, in order to return to the original simplicity of primary education. In particular the study of geometry itself, which could give the teaching its ''secondary'' tint, was removed: its practical applications (land surveying, leveling, linear drawing) remained, but as optional subjects which were only offered to certain second-or third-year student-teachers, and taught without any theoretical background.
It was only with the appointment of Victor Duruy as the Minister of Public Instruction in 1863 that teacher training, and more broadly primary education, was once again the subject of a liberal policy. Essentially for economic reasons, Duruy wanted to develop working-class education. Geometry was reintroduced into the curriculum of the e´coles normales. As proposed in the 1847 ministerial commission report (see above), geometry teaching started from the first year, in parallel with that of arithmetic. Furthermore, the e´coles normales headmasters were given a certain amount of freedom to widen the teaching provided in relation to local needs; and some did not hesitate to use it to fill out their programs, notably by including elements of algebra. The establishment of the Third Republic (1870) and the accession of Jules Ferry as Minister of Public Instruction in 1879 marked a period of relative stability, which would last until the beginning of the Second World War. For the new government, the training of teachers constituted ''the matrix place of the educational tool in making the idea of the republican nation progress'' (Grandière 2006, p. 128). 35 The subjects taught were reinforced: not only were geometry classes increased, but algebra, officially part of the program, also became a legitimate subject and part of the teacher-training. In the e´coles normales for young women, whose creation in each de´partement became mandatory in 1879, mathematics training progressively came into line with that of the male student-teachers, so that by 1920 they differed in only a few details. During the same time, the creation in the early 1880s of two e´coles normales supe´rieures d'enseignement primaire (higher primary teacher training colleges), one for young men, the other for young women, crowned the whole system of primary education. These two schools aimed to provide the e´coles normales primaires with a homogenous body of teachers (specialized in science or in arts and humanities: a certain versatility remained) and recruited from the student-teachers' workforce. While the Third Republic was attached to reinforcing the training of primary teachers (male and female), it nonetheless remained committed to building a true 'primary' culture to ensure the education of the children of the people. This primary culture differed, therefore, from the scholarly culture of secondary education (Assude and Gispert 2003;d'Enfert 2006), by giving to mathematics teaching a practical and inductive character rather than an abstract and deductive one, and by privileging practical applications.
To conclude, as prospects for further research, I would like to mention the special interest in undertaking comparative historical studies on the mathematical training of primary school teachers in European countries during the first half of the nineteenth century. 36 In this respect, France and Prussia could be fruitfully compared. Despite the differences between their political systems, these two countries showed some similarities during the 1830-1848 period: both their school systems were based on educational duality and on the distinction between ''lower primary education'' and ''higher primary education''. It would be promising to differentiate this duality and this distinction, 37 as well as the role of mathematics education-and in particular of geometry-in order to maintain them. In the same way, it would be interesting to compare the development of primary teacher training colleges in France and in Prussia, the recruitment of mathematics teachertrainers, the volume and focus of the courses in mathematics (specially in geometry) for teacher training, and to examine the respective roles of the central state and of local authorities according to the political context of each country.