Christine Chambris

Jana Visnovska

On the history of units in French elementary school arithmetic: The case of proportionality

Keywords: Mathematics education, Proportionality and ratio, Unit, Concrete and abstract numbers, Textbook analysis, Didactic transposition

The measurement of magnitudes was at the foundation of numbers and calculation in academic mathematics until the 19th century. It provided units and the concrete and abstract numbers that formed the basis of school arithmetic up to the mid-20th century in France. Our analysis of changes in teaching resources for proportionality (late 19th to the early 21st century) documents how the disappearance of magnitudes in academic knowledge was followed by the loss of the differentiation of the conceptual complexity of mathematical ideas related to proportionality. These changes made teaching and learning about proportionality considerably more difficult, and we later witness their gradual, but not yet systematic, reversal.

Résumé

La mesure des grandeurs a été aux fondements des nombres et du calcul dans les mathématiques académiques jusqu'au XIXe siècle. Elle a fourni les unités et les nombres concrets et abstraits qui ont constitué la base de l'arithmétique scolaire jusqu'au milieu du XXe siècle en France. Notre analyse de l'évolution des ressources pour l'enseignement de la proportionnalité (de la fin du XIXe siècle au début du XXIe siècle) documente la façon dont la disparition des grandeurs dans les mathématiques académiques a été suivie par une perte de différenciation de la complexité conceptuelle des notions mathématiques liées à la proportionnalité dans les ressources pour l'enseignement. Ces évolutions ont rendu l'enseignement de la proportionnalité considérablement plus difficile, nous assistons par la suite à un retour progressif mais non systématique. MSC 97-03; 97F10 ; 97F70; 97F80

Introduction

We explore how one characteristic of the "concrete versus abstract numbers" divide played out in French elementary mathematics classrooms. Drawing on the Anthropological Theory of the Didactic (ATD, [START_REF] Chevallard | Introducing the anthropological theory of the didactic: an attempt at a principled approach[END_REF], we tell the story of the "disappearance" of units from mathematics education guidelines and resources at the turn of the New Math reform and explore how this happened. We clarify the role units played in mathematics teaching prior to their "disappearance" in the development of number, show how their role changed, and question their more recent "return". In doing so, we suggest that the awareness of the role units play in meaning-making, also expressed in the concrete and abstract numbers distinction, has consequences for how we construct syllabi and how students get to understand mathematics.

The focus of our analysis is the content domain of proportionality. We clarify how the choices made at different periods would have impacted the accessibility of the mathematical ideas for students. We preface our analyses by aligning-terminologically and conceptually-with discussions in the domains of measurement and ratio in the mathematics education research literature.

Before we embark on this journey, let us evoke the broader social and historical mathematical context within which the curricular changes we document occurred. Mathematics research has a long and acclaimed history in France. The list of household-name mathematicians is, we believe, unusually long: from Descartes, Fermat, Pascal through Fourier, Cauchy, Laplace, Lagrange to Galois, Poincaré, Mandelbrot and Grothendieck, the list could go on. It is not surprising that French academics embarked on the decades long project of systematizing and unifying mathematics in the late 1930s, using the collective pseudonym of Nicolas Bourbaki. Bourbaki members, especially J. Dieudonné, were active in the preparation of the New Math reform in France and abroad.

Of additional interest is the role that France played in the history of the standardization of measurement units. The new system leveraged the mathematical power of base-10 notation, gave rise to the extended use of decimal numbers, and above all, presented measurement as an eloquent, polished, and simple collection of regularities. This would have shaped which aspects of measurement were highlighted in school syllabi and how they interacted with the remaining mathematical content.

Concrete and abstract numbers

The terms concrete and abstract numbers explored in this special issue are apparently no longer used in the teaching of mathematics in France. Yet, this terminology existed in mathematical treatises used for the teaching of arithmetic during the 18th and 19th centuries. The first chapter of Bezout's treatise (1821) as well as the notes by Reynaud related to Bezout's text (hereafter Reynaud's treatise, Reynaud, 1821)1 introduce concrete and abstract numbers within the introductory sections, among explanations of the meanings of quantities, numeration (number names and place value in base ten), number, and operations. Even though Bezout includes both whole and decimal numbers and Reynaud only whole numbers, their approaches to concrete and abstract numbers are quite similar. The first chapter of Reynaud's notes, titled Numeration, Addition, Subtraction, Multiplication, and Division of Whole Numbers, starts with a section titled The Purpose of Arithmetic:

[1]2 1. Whatever is capable of increase or diminution is called quantity. When one thinks about the nature of quantities, one feels it would be impossible to have an exact idea of values of quantities of the same kind without considering one of these magnitude values as a term of comparison; this magnitude value is called unit; combining several units of the same magnitude value forms a number. The manner of forming, expressing, and writing numbers is the object of numeration; and the science which teaches how to perform operations upon numbers is called Arithmetic. (Reynaud, 1821, 1, italics in original.) Reynaud (and Bezout) introduced number as an accumulation of units, where units were units of quantities. Reynaud further specified [2] 3. A number is abstract or concrete, depending on whether the nature of its units is considered. Thus, 3 and 5-times are abstract numbers; 3 toises3 and 5 leagues are concrete numbers. Concrete numbers formed with measurement units of several magnitude values, such as 5 toises 3 feet 4 inches, are said to be complex numbers [...] In all questions pertaining to Arithmetic, as the nature of the units of the result is known in advance, one only has to find their number; this leads to operate on abstract numbers (Reynaud, 1821, 6, italics in original).

Both "3 toises" and "3" were numbers. The former was an accumulation of units of "1 toise", where the nature of the unit (i.e., toise) was specified. The latter was an accumulation of units of 1, where the nature of the unit was not considered or specified. Abstract number thus allowed to consider units of a different nature -the unit could be "forgotten" but it could also be treated as a "variable" (i.e., 5-times). The dialectic between concrete and abstract numbers enabled to give an explanation for operations and and their execution. Concrete numbers contributed to defining, explaining, and justifying required operations, whereas calculations were mostly conducted with abstract numbers. The usefulness of the distinction between concrete and abstract numbers can be further illustrated in how the definition of multiplication was constructed:

[3] 6. The purpose of the MULTIPLICATION is to calculate a number which is called a PRODUCT. The latter is formed with a known number, called a MULTIPLICAND, in a similar way as a given number-called MULTIPLIER-is formed with the unit. Thus, to obtain the product, one has to operate on the multiplicand the same operations one would have operated on the unit to form the multiplier. [...] For instance, in order to multiply 5 by 3, one can observe that the multiplier is formed with three times the unit. The product then must be formed with three times the multiplicand 5; this product is thus 5 plus 5 plus 5 or 15. [...] The multiplier is always an abstract number because it indicates how many times the multiplicand has to be taken.

The product is of the same nature as the multiplicand; indeed, it expresses the addition of several numbers that are equal to the multiplicand (Reynaud, 1821, 10, emphasis This definition illustrates how an abstract number (here a multiplier) was purposefully imagined as a specific accumulation of a unit, which is not specified. Another quantity (expressed as an abstract or a concrete number) could then be substituted for this unspecified unit, and accumulated in the same way, resulting in the product of the multiplication.4

Units

Quantities, magnitudes, measurement units

Discussions related to numbers as a means of understanding situations and phenomena often rely on terms such as quantities, sizes, attributes, magnitudes, and units for their meaning. Yet, these words have been used differently by various authors, and demarcations of their meaning have been created for different purposes. [START_REF]Didactical Phenomenology of Mathematical Structures[END_REF], [START_REF] Passelaigue | Schoolteacher trainees' difficulties about the concepts of attribute and measurement[END_REF], and the VIM 5 (JCGM, 2012), all introduced different options for uses of measurement terminology. Given our focus on mathematical teaching and learning, we lean towards terminology that allows for making visible the distinctions that are relevant to supporting students' learning of measurement. We follow [START_REF]Didactical Phenomenology of Mathematical Structures[END_REF] in noting that the processes by which mathematics as we know it today came to be-the processes of mathematizing the world-can powerfully shape mathematical learning.

Freudenthal distinguishes four terms: quantity, magnitude, magnitude value (or value of magnitude), and measuring number. He uses magnitude to refer to an attribute of interest in a quantity of something. Examples of magnitudes include length and volume (as attributes of interest in a quantity of a wooden plank), numerosity, or sound intensity. Magnitude value then refers to a magnitude that is expressed as a property of a given thing. For example, the Earth's equator has a specific length, and the Moon has a specific volume, whether or not these magnitude values have already been measured (or quantified), or compared to another magnitude value of the same kind.

Magnitudes have long been at the core of numbers and calculations in mathematics [START_REF] Bourbaki | Eléments d'histoire des mathématiques[END_REF]. They disappeared in the second half of the 19th century, at the time of the arithmetization of analysis and of the modern axiomatizations that followed. At present time many mathematicians are no longer interested in magnitudes, and no unique mathematical definition of magnitude as a structure has been developed by the mathematics community. In spite of this, several structure definitions for magnitude can be found, often sharing a few starting points:  a magnitude is defined as a set of magnitude values,  two magnitude values can be added,  addition is associative and commutative,  the set is ordered, and  a < b if and only if there exists c, such that b = a + c.

The last property ascertains that (1) a magnitude is something which is growing when being added; and (2) subtraction is defined, and enables to "fill a gap" when one magnitude value is larger than another.

As in the the discussed treatises, Freudenthal used the term concrete number for magnitude values that have been quantified and expressed as a measurement. Specifically, when the length of Earth's equator is measured as 40075 km, this constitutes a concrete number, as the nature of the unit used is specified. To establish a concrete number, one needs to consider a unit of measurement (km), and express the relative size of the magnitude value (the length of Earth's equator) with respect to this unit (the Earth's equator is 40075 times as long as6 the kilometer).

The magnitude value is symbolically expressed with a numeral (i.e., the measuring number) and a unit name (e.g., cm). 7 A sophisticated understanding of a concrete number thus comprises the comparison between the magnitude value, and the measurement unit (see Figure 1). Figure 1. A given magnitude value, the length of a pale stick, is compared to a given unit of length, the dark measuring rod. The stick is the measuring number-times (3-times) as long as the unit.

Lastly, when working with values of the same magnitude, like in the case of the equator length and a kilometer, Freudenthal refers to these values as being from the same system. A specification of this kind becomes important when distinguishing ways in which a ratio can be constructed. Given that the word system has many different uses, we will follow [START_REF] Lamon | Rational numbers and proportional reasoning: toward a theoretical framework for research[END_REF] and refer to values of the same magnitude as being from the same "measure space". In historical sources, values of the same magnitude are often described as being "of the same kind" (e.g., Euler, 1774; Reynaud, 1821).

How units relate to ratio

Reynaud's approach to magnitudes enables one to consider any magnitude value as a unit, capable of generating numbers by being accumulated without restriction. Moreover, for any unit u of a given magnitude, and a given number a, it enables to consider other units, v and v', such that8

1 [v] = a [u], and a [v'] = 1 [u]
(for clarity, when we draw attention to a specific magnitude value l being considered as a unit, we symbolize it with brackets [l]). In such an instance, we refer to unit u as a simple unit and to units v and as units related to unit u (see also [START_REF] Chambris | Raisons d'être des grandeurs. Le cas de l'arithmétique à l'école élémentaire[END_REF]. Using the notions of simple and related units, Reynaud's definition of multiplication (section 2 above) specifies that (1) the multiplier is a given abstract number that indicates a number of unspecified units to be taken; (2) the multiplicand is conceived of as a unit to be taken, formed with a given number of simple units, it is thus a related unit; (3) the product is to be expressed as a number of simple units. In other words, with u as the simple unit, the multiplier b indicates the number of related units a u (where a u is the multiplicand) needed to form the simple units of the product b × (a

[u]) = b [a u] = P [u].
This equality means-although this is not explicitly stated in the treatises-that the magnitude value of the product (expressed in the simple units of the multiplicand), is the same as the magnitude value of b units the size of the multiplicand [START_REF] Chambris | Raisons d'être des grandeurs. Le cas de l'arithmétique à l'école élémentaire[END_REF].

For the purposes of this paper, it is important to note that as a consequence of this definition, the idea of ratio is congruent with both the idea of measurement b[a u] : [a u] = b, and of division P [u] : a [u] = b.

Ratio: internal and external

The ratio that arises as a consequence of multiplication expressed in the previous section is constructed within the same measure space, or the same system. As [START_REF]Didactical Phenomenology of Mathematical Structures[END_REF] phenomenological analysis highlights, comparisons of ratios across systems give rise to The reader will note that "1/3-times longer" is not mathematically coherent, and "3-times shorter" is usually confusing to students as the process by which the shorter length is to be created is not specified. 7 A magnitude value can also be symbolically expressed with a variable, such as the equator length l.

another type of ratio, another type of magnitude, and another type of unit. Taking the example of uniform motion, from the postulate of the uniformity of motion "in equal times equal distances are covered" (182), Freudenthal established first "in a times the time, a times the distance is covered" (a being successively n, 1/n, k/n). The relation is then extended to any positive real number through continuity of motion. Freudenthal then distinguished between internal and external ratio.

We designate ratios formed within a system as internal to distinguish them from the external ones that are discussed below.

If t 1 , t 2 are times and s 1 , s 2 corresponding paths, the postulate of uniformity says s 1 : s 2 = t 1 : t 2 .

If we are tempted to interchange the middle terms, we get s 1 : t 1 = s 2 : t 2 again the equality of two ratios, albeit ratios of path to time.

We designate ratios between two systems as external. The uniformity of the motion is now expressed by the postulate: the ratio "path to time" is constant.

Ratios can also be interpreted as quotients. In this interpretation, the internal ratio is a number, the external ratio is a magnitude, that is, in the present case of uniform motion, the quotient of path and time: speed (183, italics in original).

Understanding external ratio quantitatively requires the recognition of a new type of unitsquotient unitswhich imply a new way of mathematizing a magnitude (magnitudes of this type are referred to as intensive magnitudes). In practical mathematics (e.g., finance), external ratios of magnitudes from different measure spaces (i.e., magnitudes of different kinds) were included in procedures for calculations, even when the resulting new magnitudes were not considered or defined. However, as [START_REF]Didactical Phenomenology of Mathematical Structures[END_REF] pointed out, the use of external ratios continued to be rejected in sciences and the discipline of mathematics. As an example, Euler's Elements of Algebra (1774) specified that ratio [START_REF]il ne peut être question que de quantités d'une même espèce[END_REF] "can only be a matter of magnitude values of the same kind" (308). It was not until 1968 that a mathematical model for calculations with measurement units was proposed [START_REF] Whitney | The mathematics of physical quantities: Part II: quantity structures and dimensional analysis[END_REF].

From a phenomenological perspective, external ratio is first considered, followed by recognition that specific cases of external ratio (e.g., 60km/1h and 120km/2h) are equivalent in a proportional situation [START_REF] Cortina | Instructional design in ratio[END_REF]. These equivalent ratios represent a constant external ratio, or what [START_REF] Lamon | Rational numbers and proportional reasoning: toward a theoretical framework for research[END_REF] calls a rate. An additional conceptual step is then needed to recognise that rates constitude a new magnitude, and this new magnitude provides the meaning for a quotient unit (e.g., km/h).

During the periods under study, we illustrate tensions related to the inclusion of units in external ratio situations, and how ratios were removed altogether when proportionality was introduced through rate.

Methodology

The assumptions about how curricular phenomena develop shaped our methodology, including the data we analyzed, and the clues we looked for in our data. We adopted those assumptions from ATD [START_REF] Chevallard | Concepts fondamentaux de la didactique: perspectives apportées par une approche anthropologique[END_REF][START_REF] Chevallard | Introducing the anthropological theory of the didactic: an attempt at a principled approach[END_REF], and viewed teaching and learning as anthropological phenomena. The ATD places didactic issues at the institutional level (beyond the characteristics of individuals working within a given institution) and views teaching and learning practices as shaped by institutions (e.g., classrooms, education systems, or societies, at a given place and time).

Two of ATD's central ideas are the key to this analysis: the process of didactic transposition, and the notion of praxeology. The theory of didactic transposition [START_REF] Chevallard | Didactic transposition in mathematics education[END_REF], developed in the early 1980s, underlines the institutional relativity of knowledge (see Figure 2). Knowledge produced by scholarly institutions was reconstructed in the transposition work (e.g., [START_REF] Wijayanti | The evolution of the knowledge to be taught through educational reforms: the case of proportionality[END_REF] by educational agents (including researchers in mathematics and mathematics education, curriculum developers, policy makers) in order "to be taught", before being prepared and taught by teachers, and learned by students. ATD postulates that any human activity (notably that of mathematics) in a given institution can be described in terms of praxeologies (e.g., [START_REF] Bosch | Dialogue between theories interpreted as research praxeologies: the case of APOS and the ATD[END_REF]. The notion of praxeology indicates that practice (praxisknow-how) and the discourse (logos) about practice necessarily work hand-in-hand. There are four components in a praxeology: a type of tasks (a set of similar problems); a technique (a way of doing tasks of the same type); a technology that justifies the technique; and a theory that legitimates the technology, enables to interpret techniques, and gives meaning to tasks. The first two components form the praxis, the last two form the logos.

We attended to the presence and absence of units across different components of the praxeologies developed within the educational system to specify knowledge "to be taught". For example, as magnitudes are at the heart of ratio tasks, one might expect measurement units to appear in tasks given to students during any period. We explored how the magnitudes involved shaped the units within tasks, including whether quotient units (e.g., km/h) were present. We also analyzed the roles that units played, if any, within the theory component of the praxeology. For instance, when the treatises of Reynaud (and Bezout) were acknowledged as reference texts for knowledge to be taught, units of magnitudes were identifiable in the theory component. Moreover, both related and simple units appeared in the definitions of multiplication and internal ratio, and measurement units appeared in the definition of concrete numbers where they indicated the magnitudes at hand. Finally, we attended to how techniques and technologies were impacted by the presence or absence of units in the theory component. We particularly examined whether units were included in calculations (techniques) and whether units and magnitudes played a role in the justifications of calculations (technologies).

Analyzed texts

We analyzed the more recent syllabi9 and older reference texts-treatises [START_REF] Neyret | Contraintes et déterminations des processus de formation des enseignants[END_REF]. Both treatises and syllabi are among the textual documents that organize mathematics education by outlining what needs to be taught, and to some extent how to teach it. Treatises were considered to be a conceptual reference for teaching different topics: Their comprehensive presentation of topics and specific explanations informed the structure of textbooks and what was to be taught. The treatises we analyzed have been used for educating elementary teachers [START_REF] Neyret | Contraintes et déterminations des processus de formation des enseignants[END_REF] though they were initially designed for another purpose (educating military personnel or secondary teachers). Based on the treatises, subject syllabi were later produced to coordinate the work of a growing number of teachers. They were accompanied by materials for classroom use, including textbooks and exercise books, where the user was primarily the young learner. Treatises were a product of the transposition process, and, in praxeological terms, they developed the theory component of mathematics to be taught.

In the analysis of the knowledge to be taught, we relied on syllabi for identification of this knowledge, and on textbooks and teaching guides for additional detail. Syllabi often provide traces of scholarly knowledge that is at the origin of the didactic transposition process, as do teachers' resources (e.g., teaching guides). Technologies can be identified in the explanations, proofs, definitions, and mathematical properties that are given in solved examples. The solved examples also embed techniques, the "practical" steps for solving problems. Types of tasks are often indicated by a generic description (e.g., direct rule of three) and by solved examples. They are evident in textbooks where series of exercises occur for each type of tasks.

We focused our analysis on grades 4 and 5 (presently the final years of French elementary school), when the teaching of ratio starts. When different grade levels were analyzed for elaboration, we specifically indicate this in the text.

Historical survey of French curricular documents of 1882 to 2007

Studies on teaching numbers and arithmetic in the French education system (e.g., [START_REF] Bronner | Étude didactique des nombres réels: idécimalité et racine carrée[END_REF][START_REF] Neyret | Contraintes et déterminations des processus de formation des enseignants[END_REF] suggested that a long period of relative stability was disrupted by syllabus changes in the late 1940s, and that magnitudes disappeared from the French elementary and secondary teaching of number and operation in the period of the New Math . However, across all these periods, students in grades 4 and 5 continued to learn to solve proportionality word problems with magnitudes. What were the roles of magnitudes and units in the French elementary syllabi during these periods and how were the changes orchestrated? How did the changes impact the teaching and learning of proportionality?

To understand the role of magnitudes and units in the techniques and technologies used in the teaching of ratio in France, we build on two previously published bodies of work. [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF] used ATD to analyze French syllabi and textbooks for grades 4 to 9. She focused her analysis of textbooks on certain types of problems-simple and direct rule of three-and described the techniques and technologies taught. In her work, units were not in the forefront of the analysis. The first author used ATD to characterize changes in how the role of magnitudes and units was specified in both scholarly knowledge and knowledge to be taught in elementary school [START_REF] Chambris | Relations entre grandeurs, nombres et opérations dans les mathématiques de l'école primaire au 20e siècle: théories et écologie[END_REF][START_REF] Chambris | Raisons d'être des grandeurs. Le cas de l'arithmétique à l'école élémentaire[END_REF]. Coordinating the two analyses allowed us to (1) interpret the evolution of the teaching of proportionality with regard to the broader role of magnitudes and units in the syllabus, and (2) discuss the role of internal and external ratios in the teaching of proportionality. [START_REF] Harlé | L'image du nombre dans les manuels d'arithmétique de l'enseignement primaire au début du XXème siècle, Fragments d'histoire des mathématiques[END_REF] and [START_REF] Neyret | Contraintes et déterminations des processus de formation des enseignants[END_REF] indicated that both the Reynaud (1821) and [START_REF] Bezout | Traité d'arithmétique à l'usage de la marine et de l'artillerie par Bezout, avec des notes et des tables de logarithmes par[END_REF] treatises were used as reference texts for knowledge to be taught (see Figure 2) for syllabi, school textbooks, and for teacher education, but Reynaud's book gradually replaced Bezout's in teacher education at the turn of the 20th century. While the treatises share a general approach to magnitudes and the four operations, they differ in introducing the notion of ratio: whereas Bezout's approach was based on the theory of ratios and proportion, Reynaud's approach, which prevailed, was based on a structurally simpler theory of multiplication and division.

1882-1945: Concrete and abstract numbers during the period of internal ratios

Throughout this period, magnitudes expressed as units were included in the documents that were produced to organize arithmetic learning. In 1882, in grade one, students learned [START_REF]Addition et soustraction sur les nombres concrets et ne dépassant pas la première centaine[END_REF] "addition and subtraction on concrete numbers and not exceeding the first hundred" (Syll1882, italics are ours). The subsequent syllabus (for the 2nd grade in 1923) included, in the same paragraphs, discrete and continuous magnitudes. Indeed, both discrete and continuous measurements were involved in the treatment of whole numbers, operations, and place value topics. Moreover, from 1923 on, place value was clearly connected with the metric system. The 4th grade 1923 syllabus indicated that: [START_REF]Les élèves comprendront ce qu'est un dixième de mètre, un dixième de gramme, avant de comprendre ce qu'est un dixième d'unité[END_REF] "Students will understand what is a tenth of a meter, a tenth of a gram, before they understand what is a tenth of a unit" (Syll1923, italics are ours). The role of magnitudes and units seemed not to have changed: measurement of magnitudes led to numbers and operations, and numbers and operations were taught using related and measurement units.

In ratio and proportion in grades 4 and 5, simple calculation of "rule of three" problems appeared to be the goal. Hersant drew attention to a teacher's guide that interpreted the 1882 Syllabus in the following way: [START_REF]Ne rejetons pas trop loin la règle de trois : il y a avantage à rompre les enfants à cet exercice de raisonnement qui permet de résoudre au moyen des quatre opérations seulement une foule de[END_REF] Let us not delay too much the teaching of the rule of three: it is advantageous for children to get used to this reasoning exercise, which allows them to solve a number of practical questions through the four operations alone. There is no need to wait for the study of ratios and proportions (Mutelet and Dangueuger, 1910, 2nd edition. Quoted by Hersant, 2005, 11).

This excerpt refers to the reduction to the unit method. In the six textbooks Hersant analyzed, the justification of this method was mainly based on the comparison of the variation of magnitudes that embedded implicitly internal ratios. For instance, [8] "Two magnitudes are directly proportional if when one of them becomes 2, 3, etc. times larger or smaller, then the other becomes 2, 3, etc. times larger or smaller" (Lemoine, 1913, 124). The method was very similar to the one presented by Reynaud: it was based on multiplication and division, not on calculation with ratios. Yet, a major difference was that the justification was mostly explicit. As in Bezout's book, it was based on the comparison of the variation of magnitudes, and used the language of proportions: [9] "Thus, the price of a commodity is directly proportional to its length, or to its surface, or to its weight, or to its volume" (Lemoine, 1913, 124). The "reduction to the unit" technique was then made of three steps written in a spatial pattern in three lines. A justification, using expressions "times less", "times more", was often included in solved examples in textbooks, like in this worker salary calculation problem = 70 francs (Lemoine, 1913, 125).

Though the reduction to the unit method was the only technique considered legitimate in the educational system up to 1923, the method was contested in many textbooks, and alternative techniques were offered [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF]. These alternatives only involved internal ratios and attended to relationships within the two spaces of measurement. First, the method was challenged because its full algorithmization sometimes led to heavy calculations that could be avoided. In most textbooks, alternative techniques for mental calculation used fractional internal ratios, such as [START_REF]'il serait impossible de prendre une idée exacte des grandeurs des quantités de même espèce, si l'on ne choisissait pas parmi elles une certaine quantité qui pût leur servir de terme de comparaison ; cette quantité se nomme unité ; l'assemblage de plusieurs unités de même grandeur compose un nombre. La manière de former les nombres, de les énoncer et de les écrire[END_REF] "the price of 12 kg is equal to 4 times the price of 3 kg (10fr,50): 10fr,50 . 4 = 42 fr" (Mortreux and Mortreux, 1910, 357).

The second point of contention concerning the reduction to the unit method was that "absurd" magnitude values appeared as intermediate results in calculations. We illustrate this with the example of a problem, where 13 workers do 273 m of work, and the task is to find out how many workers would do 420 m of work in the same amount of time. The first step in the reduction to the unit method is: [12] "1 m is done by

13 273
worker" (Mortreux and Mortreux, 1910, 358). To this, the authors add [13] "We are making an absurd reasoning (sic)" (Mortreux and Mortreux, 1910, 358, italics in original). Alternative techniques that aimed to avoid the "absurd" intermediate results involved the notion of unit value.

[14] The required number of workers is equal to the total length (420 m) divided by the length made by 1 worker.

Length made by 1 worker: When [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF] quoted the same example solution (from a 1933 new edition of the textbook for grade 6 of the same series), she suggested that this technique was less algorithmic than reduction to the unit, and that it used coefficients (i.e., external ratios). Given the historical situation of the grade 4 and 5 textbook, we suggest that an alternative explanation is more plausible. The convention by which the measurement units were not included in calculations, but they were used in explanations, suggests that the unit value, here the "length made by 1 worker", was thought of as a length measured in meters. The "13" in is not a coefficient. We propose that this technique was based-at the technology level-on multiplication and division and it did not involve an external ratio.

From the beginning of the period, a few 6 to 7th grade textbooks stated that when magnitudes are proportional the external ratios are equal. While some textbooks introduced calculation with multiplication or division of magnitude values, other textbooks rejected ratios of magnitude values of different kinds [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF]. Our analyses lead us to suggest that, in order to support students in solving the types of problems that were expected of them, techniques involving external ratios must have been taught, even in grades 4 to 5. Indeed, syllabi from 1882 (Syll1882) indicated that problems that included interest rates were studied in these grades, and problems including density were added in the sixth and seventh years. Both notions of interest rate and relative density were constructed as external ratios of values of two magnitudes of the same kind. For instance, problems of profit calculation involved multiplicative comparison of the money gained and the money paid. Because the two measurement spaces involved were of the same kind, the resulting interest rate, often expressed as a percentage, could be treated as an abstract number (a quotient of two monetary magnitude values).

The analysis of the techniques proposed for solving interest and density problems suggests that inclusion of external ratios was a complex issue. In textbooks, interest percentage was at times presented as an external constant ratio: [15] "The 6% profit represents 6 hundredths of the purchase price" (Lemoine, 1913, 141). However, it was more often defined as 6 fr for each 100 fr, and the main method taught to solve these problems was an adaptation of reduction to the unit, explained as follows:

[16] A house is sold with a profit of 6%. Knowing that this profit is 300 francs, what is the purchase price of the house? Solution:

A profit of 6 francs comes from a purchase of 100 francs; a profit of 1 franc comes from a purchase of 100 fr 6;

and a profit of 300 francs comes from a purchase of 100 fr×300 6 = 5000 francs (Lemoine, 1913, 141).

Another solution was given, also based on internal ratios [17] "The purchase price would include 100 francs as many times as there are 6 francs in the profit of 300 francs, that is 100 fr × 300 6 = 5000 francs" (Lemoine, 1913, 141). Textbooks that presented solutions based on external ratio relied on fraction knowledge that was developed in previous chapters and led to the direct calculation of the reciprocal of 6 hundredths, of 300 francs. We note that the magnitudes in the proportion (money gained and money paid) were not always mentioned in the solution for this type of problems. Interestingly, some textbooks did not connect these problems with the rule of three, and suggested a solution technique based on a formula, without providing an explanation for why it worked.

The last case of the use of external ratios in the 1938 syllabus, for grades 6 and 7, did not bring more clarity. The syllabus introduced specific "quotients" (Syll1938) such as price per capacity unit, or mass per unit of volume. We include this discussion here to illustrate a possible departure from previous uses of the unit value technique, and because the following period will see these types of tasks and techniques used in grades 4 through 5. The syllabus specified that quotients could be considered as unit values, and suggested solving rule of three problems with them. For this purpose, a formula was given, [18] "total value = unit value × number of units" (Syll1938), but the proposed solution did not include measurement units of the components of the multiplication. 10 It is thus unclear whether the intended "unit value" in the "price per capacity" example would be "price" (changes in which needed to be compared to corresponding changes in capacity, that is, internal ratio), or whether "price per capacity" was considered an admissible composite magnitude for a "unit value" in this situation.

The above question is substantiated by the fact that, in examples of calculations with decimals, the same syllabus includes measurement units in calculations, when justifying the rule of multiplication by a decimal: [19] "(3.50 fr per l) × (7.25 l) could be replaced by (0.035 fr per cl) × (725 cl)". 11 Here, "3.50 fr per l" and "0.035 fr per cl" appear to be treated as the 10 There were no quotient units in calculations provided in reference texts, i.e., the treatises. Such calculations seem to have been introduced in the 1938 syllabus, aiming at enriching the theoretical environment of arithmetic, without drastically changing it. 11 In France, a comma is used in decimal notation (instead of a decimal point). In this paper, we use the English notation in translations and keep the original notation in the originals. same intensive quantity, "price per capacity unit", expressed with two different units ("fr per l" and "fr per cl").12 In light of this example it becomes clear that, with the introduction of the formula "total value = unit value × number of units", the unit value technique came to involve two different meanings, depending on whether the "unit value" was viewed to be of a simple or of a quotient magnitude. In the former case ("length made by 1 worker"), the unit value, the unit, and the unit count were all fixed. In the latter case, while the magnitude values of both the quotient "unit value" and the corresponding "number of units" remained fixed, the measuring number that expressed the "number of units" could vary: in our example, it changed from 7.25 to 725.

The difference in the two possible uses of the unit value technique sheds light on the complexities that are introduced with intensive quantities (i.e., external ratios of values from different measure spaces): the work with abstract numbers, where the units are not specified, is no longer meaningful. This introduction forces the use of magnitude values, and thus measurement units, in calculations.

Notions of interest rate, density, and quotients justified the introduction of specific techniques in calculation, but some of the techniques in the textbooks were devoid of underlying technologies. This shows that the introduction of work with external ratios of different magnitudes had been tentative at best. While the quotients represented new types of magnitudes, and required that calculations consider magnitude values (i.e., work with concrete numbers), they were not consistently treated as such. The notion of the constant external ratio, and its relation to reasoning with internal ratios in proportional situations was not explicitly addressed during this period in grades 4 and 5 and was only rarely addressed in grades 6 and 7.

1945-1970: Departure from internal ratios in grades 4 to 5

Lebesgue's book La mesure des grandeurs (Lebesgue, 1975 (posthumous)), published serially from 1931 on, progressively became a reference book for elementary teacher education [START_REF] Neyret | Contraintes et déterminations des processus de formation des enseignants[END_REF]. A disruption with regard to arithmetic was the inclusion of the set of real numbers, whereas prior treatises were limited to fractions. It included a construction of real numbers based on measurement with sequences of decimal length subunits, without including either a theory of fractions or Euclidean ratios. The replacement of fractions with decimals was motivated by an attempt to avoid definitions by abstraction [START_REF] Chambris | Relations entre grandeurs, nombres et opérations dans les mathématiques de l'école primaire au 20e siècle: théories et écologie[END_REF] that were, at the time, mathematically problematic [START_REF] Griesel | Reform of the construction of the number system with reference to Gottlob Frege[END_REF][START_REF] Otte | Mathematical history, philosophy and education[END_REF]. Despite a different presentation compared to prior treatises, multiplication was again based on the change of units of measurement and general rules for calculations with real numbers were developed. This resulted in the algebraization of numbers. Neither internal, nor external ratios were used in the presentation of proportionality, but Lebesgue proposed a rationale for developing proportionality from a postulate of uniformity to the existence of a rate.

In the 1945 syllabus, the status of magnitudes kept changing. Concrete and abstract numbers remained and quotient units appeared. Calculation with magnitudes was contested, somewhat inconsistently. In line with Lebesgue's construction of real numbers, fractions no longer appeared as a means to develop decimals [START_REF] Chambris | Relations entre grandeurs, nombres et opérations dans les mathématiques de l'école primaire au 20e siècle: théories et écologie[END_REF]. Decimals were to be taught as concrete numbers, using the metric system (e.g., [20] "2 meters and 15 centimeters = 2.15 m", Syll1945, 246).

In 1945 through 1970, the reduction to the unit method continued to be included in the syllabus under the name rule of three where the rules developed by Lebesgue were introduced as calculation formulas. Two other methods appeared in grades 4 and 5, and the choice of methods depended implicitly on the kinds of magnitudes in the proportion. Firstly, similar to the earlier treatment of interest rates, external ratio comparison of two magnitudes of the same kind resulted in a dimensionless rate. Tasks included calculations such as [START_REF]poids de farine = poids de blé × 0[END_REF] "weight of flour = weight of wheat × 0.80" (Syll1945, 251). 13 Fractions were considered abstract multipliers and were treated as fractions of magnitudes (i.e., take four fifth of) in ratio problems.

Secondly, the unit value method, indicated in 1938 for higher grades, was to be used otherwise, and the uncertainty about the type of units to be used was solved: quotient units were introduced. This method led to the introduction of formulas, and to a certain algebraization of the technique. It increased the demands for prerequisite mathematical understandings, as younger students would have to: (1) learn formulas by heart, or (2) learn basic algebra to manipulate multiplicative equalities, or (3) work with the complexity of quotient units. In fact, the method was introduced in grades 2 and 3 for learning multiplication and division, as well as the notions of multiplicand and multiplier. This change might be interpreted as an attempt to increase the time students had for learning methods related to ratio. The introduction of external ratio techniques for early grades made justifying calculations with magnitude values even more difficult.

In textbooks, the reduction to the unit method remained in the fore, but the definition of proportional magnitudes (the comparison of the variation of magnitudes) and the justification of the method through internal ratios became increasingly less explicit [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF]. "Spatial explanations," started to replace them: [START_REF]Je cherche un prix. Je place le prix 18 F à droite[END_REF] "I'm seeking a price. I place the price 18 fr on the right" (Benhaïm and Nadaud, 1969, 109) and mental arithmetic using internal ratio declined. Many textbooks focused on calculations, such as a fraction of a magnitude value, but without distinguishing whether the calculation represented an internal or external ratio. The proportionality of the problem situation was not always pointed out. In spite of the syllabus recommendation, the unit value method was rarely included in elementary textbooks [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF].

The influence of Lebesgue's book prompted the decrease of the use of fractions (in the syllabus and textbooks). Decimals, introduced through the metric system, provided an alternative for expanding beyond whole numbers. The discourses (technologies) for proportionality also changed, where expressions such as times more, times less, and two thirds of were now avoided, and exclusively spatial and multiplicative symbolic expressions were preferred. Algebraic rules, not rooted in sense making, progressively became the only means of providing coherent justification, even though concrete numbers remained in use. The situation was ripe for removing magnitude values from mathematical consideration altogether.

1970-2002: Units prohibited in the period of ratio as function

The introduction of the reformed syllabus14 in 1970 asserted that the curriculum was not a new one but a [23] "different writing" (Syll1970, 285) of the previous one from 1945, a huge ecological and praxeological reorganization, we might add. On the surface, the New Math reform resulted in a reorganization of the topics, but these, while bearing the same names, now referred to a different scholarly knowledge. The transposition of new scholarly knowledge reached far beyond the introduction of set theory in the elementary grades, for which the syllabus is commonly known. It brought about changes in the knowledge to be taught for fractions and decimals, and had unprecedented consequences in the teaching of ratio.

Background on number: scholarly knowledge and syllabi from 1970 on

Previous syllabi included only two domains: arithmetic and geometry. The 1970 syllabus was reorganized into three domains: (a) the numerical domain which focused on numbers and operations, and was titled Elements of Mathematics, following the name of Bourbaki's series, (b) the geometrical domain titled Observation Exercises and Work on Geometric Objects, and (c) the measurement domain titled Measurement: Practical Exercises. The name indicates that measurement was linked with pragmatic needs, unlike the numerical domain which now introduced elements of a (new) system of thought.

The creation of the measurement domain resulted in a separation of discrete and continuous magnitudes. They were previously studied in parallel, providing meanings and imagery to the concept of number. In 1970, students were instead expected to learn the [START_REF]concept de nombre naturel[END_REF] "concept of natural number" (Syll1970, 286) almost entirely through counting (i.e., by quantifying discrete magnitudes) in the numerical domain, and to apply it to continuous magnitudes in the measurement domain.

Paradoxically, the birth of the measurement domain is the visible side of the disappearance of magnitudes as the conceptual roots of numbers. The syllabus writers decided to align school mathematics with the new scholarly mathematics, where numbers now came from the theory of real numbers and were based on sets. Continuous magnitudes were thus no longer required in schools for defining and teaching whole numbers, fractions, or decimals [START_REF] Chambris | Relations entre grandeurs, nombres et opérations dans les mathématiques de l'école primaire au 20e siècle: théories et écologie[END_REF].

A series of changes was triggered by this syllabus, as the education system attempted to make the new scholarly knowledge teachable. While the measurement domain remained for years to come, from 1980 on, continuous magnitudes increasingly returned to the numerical domain in tasks that aimed to support meaning making.

Against this background, the changes in the meaning of operations and the teaching of proportionality that the 1970 syllabus initiated remained central for almost 30 years.

Operations with bare numbers

The 1970 syllabus clearly differentiates between operations, and situations to which the operations correspond. [START_REF]L'étude des nombres naturels comprend celle des deux opérations fondamentales, l'addition et la multiplication, qui donnent à l'ensemble de ces nombres sa structure algébrique propre. A ces deux opérations se rattachent la soustraction, la division exacte et la division euclidienne (c'est-à-dire avec reste pouvant être différent de zéro). Il est essentiel de comprendre que l'addition, la multiplication ne portent que sur des nombres. Il est tout aussi important que les enfants reconnaissent les situations auxquelles correspondent ces opérations[END_REF] The study of natural numbers includes the study of the two fundamental operations, addition and multiplication, which give the set of these numbers its own algebraic structure.

These two operations are linked to subtraction, exact division and Euclidean division (i.e., with a remainder that may be different from zero).

It is essential to understand that the addition, the multiplication only concern numbers. It is equally important that children recognize the situations to which these operations correspond. (Syll1970, 289-290, italics are ours.) Hence, the (mathematical) scholarly knowledge seems to no longer provide justification or explanation for knowing the nature of the units of the result. [START_REF]porte sur les nombres et non sur les ensembles que ces nombres qualifient : on réunit des ensembles d'objets ; on additionne des nombres. Les phrases telles que : 8 pommes + 7 pommes = 15 pommes. n'appartiennent en fait, ni au langage mathématique, ni au langage usuel. Le langage courant utilise, en effet, des phrases telles que "lorsque j'ajoute 8 pommes aux 7 pommes qui sont dans la corbeille, la corbeille contient 15 pommes" ou même de façon plus vague "8 pommes et 7 pommes, cela fait 15 pommes[END_REF] Let it be recalled that addition (and also subtraction, multiplication, etc.) refers to numbers and not to the sets that these numbers qualify: sets of objects are brought together; numbers are added together. Sentences such as: 8 apples + 7 apples = 15 apples. do not actually belong to either mathematical or common language. Common language uses phrases such as "when I add 8 apples to the 7 apples in the basket, the basket contains 15 apples" or even more vaguely "8 apples and 7 apples, make 15 apples." Add, and are words of current language, they are not terms used in mathematical language. Conversely, the word "plus" is not usually used in everyday language to express the action of adding. In classroom practice, the two languages are mixed, but it is important to distinguish them. For example, we could write: The number of apples is: 8 + 7 = 15 and conclude: "The basket contains 15 apples" (Syll1970, 290, our italics).

This excerpt illustrates that numbers with units were no longer considered mathematical and that they were seen as objects of daily life and language. Calculation techniques, including those used for proportionality, prohibited the use of measurement units. In this period, proportionality was a relation between lists of numbers, and this allowed the removal of the remaining traces of magnitudes.

Proportionality as a numerical function

The 1970 French elementary syllabus introduced-for the first time-the term numerical relations. In the following 1980 syllabus, this term was mostly replaced by numerical functions. In 1970 through 2002, proportionality was explicitly considered as a case of numerical function (or relation), and the key purpose for its inclusion was to teach numerical functions rather than to teach proportionality.

The 1970 syllabus presented several two-row tables of numbers, without any context, with an explicit rule for number correspondence (e.g., adding 4; multiplying by 3). It defined "numerical relations", and their "reciprocal relations" (i.e., subtracting 4; dividing by 3, respectively), while the term "reciprocal" remained implicit. Based on [START_REF]situations concrètes[END_REF] "the study of concrete situations" (Syll1970, 298), proportionality was presented as follows: [START_REF]passer d'une liste à l'autre est la proportionnalité[END_REF] "When the operator is "multiply by..." or "divide by..." the correspondence that enables to move from one list to another is called proportionality" (Syll1970, 298). Three examples of concrete situations were given, including the relation between length and area: [START_REF] Syll | [END_REF] A series of rectangles is considered, one side of which, a unit being chosen, is measured by the number 4. A grid is used to determine the areas of these rectangles taking the square of side 1 as the unit of area. (...)

Length of the 2nd side

Area

We can construct the table of correspondences between the measure of the second side and the area of these rectangles. More generally, if the measure of the second side is a, the area (the unit being determined as above) is 4 a (Syll1970, 299). 2 8 3 .

5 .

4 .

.

24

. 36

a 4 a
The solutions recommended the use of a coefficient, here "multiply by 4", and numbers were intended to represent units of length and area. Additive and multiplicative properties of the multiplicative numerical relations were derived from the lists of numbers taken from the examples, then generalized with a formula.

The 1980 syllabus built on the earlier syllabus. While [START_REF]situations rencontrées en classe ou hors de la classe[END_REF][START_REF]compteur de pompe à essence[END_REF] "situations encountered in or out of the classroom ... [such as] a fuel pump meter" (Syll1980, 53) were still used to provide motivation for numerical relations, these relations were described with "numerical data" (i.e., measuring numbers), and magnitudes were used infrequently. Proportionality was presented in a specific sub-section as a characteristic property of numerical functions 𝑓: 𝑛 → 𝑛 × 𝑎, a and n being numbers, not magnitude values. Measurement units appeared only once in the section devoted to function, in a "scale" problem and its solution: [START_REF]de 6 = 2 × 3 (en cm) est de 2,5 × 3 (en km) sans qu'il soit ici nécessaire d'expliciter le coefficient de proportionnalité (en l'occurrence "l'échelle[END_REF] "2 cm on the map represents 2.5 km on the ground. What is the distance that, in the field, corresponds to 6 cm on the map?" [...]

Students can then determine that the number corresponding to 6 = 2 × 3 (in cm) is 2.5 × 3 (in km) without the need here to clarify the proportionality coefficient (in this case the "scale")15 [...] (Syll1980, 56, our italics).

A calculation method was provided to avoid the proportionality coefficient, presumably because it is easier to get, multiplicatively, from 2 to 6 than to get from 2 to 2.5. This method relied on internal ratios but it only indicated this implicitly, and measurement units were used here as a clarification of where the numbers in the calculation originated. The calculations were enacted with measuring numbers.

At this time, the functional relationship was defined through a constant multiplicative relationship between two lists of numbers (called coefficient, i.e., rate expressed without units). Over time, properties of linear functions were increasingly intended to serve as an alternative means of solving problems [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF]. In 1980 and 1995 syllabi (Syll1980, Syll1995), techniques for representing functions graphically also provided new means of solving problems [START_REF] Hersant | La proportionnalité dans l'enseignement obligatoire en France, d'hier à aujourd'hui[END_REF].

The study of [START_REF] Sokona | Aspects analytiques et aspects analogiques de la proportionnalité dans une situation de formulation[END_REF] indicates the types of reasoning students learned in this period. He followed students in grades 6 and 10 as they solved problems that involved proportional relations in a new situation. He observed that although the "coefficient" (a relation between measure spaces) appeared as the main "taught technique", students were not able to use it to solve problems in a new context. The students who were successful attended to relations of magnitudes within measure spaces, even though such relations were not the focus of teaching during this period.

From 2002: A tentative return of units in operations and proportionality

In 2002, functions had been removed from the elementary syllabus. For proportionality, the background of function remained, but it became implicit: the term function no longer appeared. The purpose of the study of proportionality was now [START_REF]étendre la reconnaissance de problèmes qui relèvent du domaine multiplicatif[END_REF] "to extend the recognition of problems that fall within the multiplicative domain" (Syll2002, 16). From 2002 on, official texts gradually returned to discussing proportionality as a relation between magnitudes.

The 2002 syllabus gave examples of expected "students' reasoning" (Syll2002, 16), which included numbers with units. The reasoning techniques included additive and multiplicative relationships within measure spaces, including reduction to the unit method (under the name Internal and external ratio situations were both used but their difference was not discussed. Indeed, the same comparison language (e.g., n times less) described both the decrease in one quantity (fruit) and comparison of different quantities (fruit and sugar).

The syllabus suggested proportional situations with average speed, or where an amount of liquid and its height in a cylindrical container were considered. These situations involved magnitudes of different kinds (e.g., length and time, volume and length). However, the ways for dealing with the magnitudes were not exemplified. All the solved examples involved two magnitudes of the same kind (e.g., mass).

In operations, the equal sign was only ever used with bare numbers. Operations with numbers with units seemed to have been employed to address pedagogical needs, but were not treated as belonging to mathematics. Coefficients remained rates without units. The official 2005 text titled Magnitudes and measurement in elementary school included partial mathematical justifications of manipulation of magnitude values and units.

[36] Since the quantities considered (lengths, areas, volumes, durations, masses) can be added, subtracted, multiplied or divided by a number, the following entries are correct and their use is recommended: Supporting the reasoning with external ratios seems to have become, again, a thorny problem for curricular developers. Indeed, in a 2007 official text for grades 6 through 9, proportionality was defined as a relation between magnitudes. External ratios were presented with units, and such a presentation was said to be expected from students in grades 8 and higher (MEN, 2007). This text also explained that prior to grade 8, it was better to only give students problems with magnitudes of the same kind. While keeping the same characterization of proportionality across the grades (i.e., with a coefficient), this clarification most likely aimed at sparing younger students having to work with quotient units. However, this was not consistent with the ongoing 2002 syllabus, which included speed.

Even though fractions were used to express magnitude values, fraction tasks that would involve internal ratio (e.g., 3/4 of a given length, 2/5 of 300 g, 7/4 as long as a pencil) were rare in the elementary syllabi. In 1970 to 2002, such tasks were studied in the function domain as a specific case of composition of numerical functions (e.g., 3/15 as: first multiply by 3, then divide by 15). In the 2002 syllabus, these tasks were moved to the section on "arithmetical knowledge of numbers". In spite of this change, the tasks with fractions conceptualized as internal ratios did not appear (with one exception, related to angles). Therefore, in the early 2000s, the state of teaching internal ratio in the context of numbers hindered offering internal ratio as an alternative for reconciling ratio problems with different kinds of magnitudes.

Discussion and conclusion

The documented changes in syllabi and textbooks may appear chaotic, or even disorganized: different techniques appear to fall out of favor at various times, only to re-appear again later.

A different picture emerges when we take the profound change in the scholarly mathematics knowledge in this historical period into account, and interpret the subsequent syllabi changes as attempts at making scholarly knowledge teachable (see Table 1).

At the beginning of our analysis, the mathematics in the reference texts for teaching was axiomatized as an idealization of reality [START_REF] Otte | Mathematical history, philosophy and education[END_REF], and numbers were generated from the measurement of magnitudes. This explicit connection between reality and mathematics also inconspicuously provided novice students with a convincing, intuitive rationale for learning numbers, including the ways of multiplicatively organizing continuous magnitudes, which became essential in proportionality.

However, the mathematicians' quest for coherence in the discipline was resolved by its initially gradual, and later determined departure from reality as a source of, or reference for, mathematical objects. While the change towards set theoretical abstraction benefited scholarly mathematics, it no longer provided a compelling reason for learning number in classrooms, and it removed previously essential reasoning tools in the form of magnitudes and units from the learner's, and the teacher's, repertoire. It soon became apparent to teachers and curriculum designers alike that the new syllabi did not well support students' learning. Units and magnitudes, which allow students to connect mathematics to reality, were reintroduced to provide both the need and meaning for mathematics.

Concerning improvements in teaching in the first period (before 1945), proportionality was initially based on the strong foundation of multiplicative relationships for magnitudes and units. Syllabus revisions introduced more sophisticated tools (external ratio and rate), when some of the complexities related to external ratios of magnitudes, intensive magnitudes, and quotient units were (even tentatively) resolved.

The project of building academic mathematics without contradictions saw the syllabus start anew. This time, it began with the most encompassing, sophisticated, but also conceptually dense notion of proportionality, the notion of rate. The less sophisticated notions (additive comparisons, and internal and external ratios), which previously derived their meaning from magnitudes and units, and provided stepping stones for the conceptual idea of rate, were left out. While the 1945 syllabus attempted to keep-at the theoretical level-a connection to magnitudes as a source of concrete numbers, the subsequent syllabus departed from the connection of numbers to magnitudes altogether.

. Having magnitudes at the center of academic mathematical work had two major advantages. First, mathematical theory itself provided the primary model for the process by which the experiential world could be mathematized. "Concrete" magnitudes, as attributes recognized in everyday objects, provided measurement units from which the first layer of mathematical abstraction-the number magnitude and number units-could be built. The difference between a concrete magnitude and the number magnitude is at the core of the distinction between concrete and abstract numbers, which, at the time, co-constituted the meaning of "doing mathematics".

The second advantage relates to the tools for reasoning multiplicatively. The treatment of units within a mathematics theory (i.e., within a magnitude, anything can be a unit) allowed for flexibility in working with numbers. Specifically, by bringing explicit attention to the process by which related units are created from simple units, it allowed one to define multiplication, fractions, and internal ratio. Because these definitions were constructive, they supported the imagery of how these mathematical notions related to learners' prior experiences. The same tools that were used to define mathematics could be deployed to make sense of it.

Across all periods, quotient units presented a problem. Dividing kilometers by minutes did not seem correct, and the realization that the relationship of distance and time constitutes a new, intensive magnitude was required. Not surprisingly, the inclusion of the external ratio was problematic. But even more problematic, we suggest, was-and continues to be-the lack of attention to the differences in the complexity of multiplicative reasoning involved in the different ways in which ratios and rates are constructed. Ratio comparisons within magnitudes, across quantities of the same magnitude, across different magnitudes, and even rates, were all treated as simply exchangeable calculation methods. Freudenthal pointed this out in 1983, stating that "since no-one today is conscious of the mental jump from internal to external ratios, nobody raises the question as to whether it could not be too big for the learner" (184).

The return to leveraging students' experiences with magnitudes is a promising start of a more viable teaching of proportionality. It allows one to develop the need and ability to reason about magnitude values quantitatively. Yet, we suggest that more attention needs to be paid to the role of units and how they make the process of mathematizing multiplicative relationships possible (cf. [START_REF] Bruyset | Mathematics as an Educational Task[END_REF]. The internal ratio might need to reappear as a tool for understanding external ratios and rates.

questions pratiques. Inutile d'attendre pour cela l'étude des rapports et des proportions. [START_REF] Mutelet | Programmes officiels des écoles élémentaires, interprétation, division, emploi du temps: à l'usage des instituteurs, institutrices et des candidats au certificat d'aptitude pédagogique[END_REF]Dangueuger, 1910, 2nd edition. Quoted by Hersant, 2005, 11) [8] Deux quantités sont directement proportionnelles lorsque l'une d'elles devenant 2, 3, etc.

fois plus grande ou plus petite, l'autre devient de même 2, 3, etc. fois plus grande ou plus petite. (Lemoine, 1913, 124) [9] Ainsi, le prix d'une marchandise est directement proportionnel ou à sa longueur, ou à sa surface, ou à son poids, ou à son volume. (Lemoine, 1913, 124) [10] (Lemoine, 1913, 125) [11] Le prix de 12 kg. est égal à 4 fois le prix de 3 kg. (10f,50)): 10f,50  4 = 42 fr (Mortreux and Mortreux, 1910, 357) [12] 1 m. sont faits par 13 273 d'ouvrier. (Mortreux and Mortreux, 1910, 358) [13] nous faisons un raisonnement absurde. (Mortreux and Mortreux, 1910, 358, italics in original) [14] Le nombre des ouvriers nécessaires est égal à la longueur totale (420 m.), divisée par la longueur que fait 1 ouvrier.

Longueur faite par 1 ouv. : = 20 ouv. (Mortreux and Mortreux, 1910, 358) [15] Le bénéfice 6% représente les 6 centièmes du prix d'achat. (Lemoine, 1913, 141) [16] Une maison est revendue avec un bénéfice de 6%. Sachant que ce bénéfice est de 300 Francs, on demande le prix d'achat de la maison. Solution. Un bénéfice de 6 francs provient d'un achat de 100 francs ; Un bénéfice de 1 franc provient d'un achat de = 5000 francs (Lemoine, 1913, 141) [18] valeur totale = valeur de l'unité × nombre d'unités (Syll1938)

[19] (3,50 fr. par l. ) × (7, 25 l. ) peut être remplacé par (0,035 fr. par cl. ) × (725 cl. ) (syll1938) Section 5.2

  in the original text).

Figure 2 .

 2 Figure 2. The process of didactic transposition. This conceptualization implies that what students learn is profoundly shaped by what is taught by their teachers, which in turn is shaped by what appears in written curricula (including syllabi and textbooks), which involves knowledge to be taught produced by educational agents through their transposition work.

[ 10 ]

 10 For 16 days, the worker receives 40 francs; For one day he will receive 16 times less, or 40 fr 16 ; And for 28 days he will receive 28 times more, or 40 fr×28 16

  workers(Mortreux and Mortreux, 1910, 358, italics ours).

  for the number of workers. Instead, it expresses the multiplicative relation between the length produced by 13 workers and what is produced by 1 worker as being the same as the multiplicative relation between the numerosity of 13 workers and one worker (i.e. 1 13 , proportionality definition as a correspondence of variations within two measure spaces). In this context, the 273 m 13

  prix d'achat se compose d'autant de fois 100 Francs qu'il y a de fois 6 francs dans le bénéfice 300 francs, c'est-à-dire 100 fr × 300 6

The first edition of Bezout's treatise was published in 1764, Reynaud's notes in the early 1800s.

The original French text can be found in Appendix 1, under the corresponding indicator "[n]". Unless stated differently, all translations are by the first author.

The French original refers to "toises" and "lieues", units of length used in France before the adoption of the metric system.

Even though this definition used whole numbers as an example of unit accumulation, it allowed for any multiplicand to be accumulated by any abstract number (considering a measurement process). The same definition was used for multiplying fractions in a later section of the treatise.

The VIM (International Vocabulary of Metrology -Basic and general concepts and associated terms) is the international reference book for metrology.

We use "z-times as long as" to express multiplicative comparisons, instead of "z-times longer", because the former expression captures the required relationship in mathematically meaningful way for any real number z.

Note that the second case postulates the existence of sub-units of u that correspond to unitary fractions. It points to situations where 10 v' = u, which establishes v' = 1/10 u.

French syllabus documents generally include an introduction, a list of topics to be taught (the syllabus in the strict sense of the word), and more or less detailed instructions that indicate how to teach each topic. In referencing syllabi, we use the notation 'Syll1923' to indicate that the reference is one of the source texts for our analysis.

The recognition of the equivalence of the quotient "unit values" (e.g. 3.50 fr per l = 0.035 fr per cl) presumes recognition of a rate.

"Weight of flower/weight of wheat" is an external ratio.

We use "reformed" here as this is how the new syllabus was known as at the time. It presented a major restructuring of how mathematics is to be taught in schools. The French mathematician André Lichnérowicz headed the ministerial commission on mathematics education in France(1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973).

In this case, the proportionality coefficient is 1.25 and the scale is 1:125 000.

m × 7 m = 28 m 2 […]156 km 2 h = 78 km/h."(MEN, 2005, 82)

Acknowledgements

We would like to thank Marianna Bosch and Jose Luis Cortina for their generous suggestions, and the special issue editors Christine Proust and Eric Vandendriessche for encouragement and direction they provided through the many versions of the manuscript.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Scholarly knowledge

Knowledge to be taught Table 1. Summary of changes in arithmetic/numerical domain of syllabi and textbooks and their relation to changes in scholarly knowledge (MU = measurement units, NU = number units, QU = quotient units, (SU/RU) units were organized as simple and related units).

AppendixA. Supplementary material [see further]

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.hm.2021.04.003.

Appendix 1. Supplementary material. Original texts in French

The following is the Supplementary material related to this article.

Section 2


L Sciacovelli

P Cinnella

X Gloerfelt

Analysis of dense gas effects in compressible turbulent channel flows

Introduction

In this work we investigate the influence of dense gas effects on compressible wallbounded turbulence. Turbulent flows of dense gases represent a research field of great importance for a wide range of applications in engineering. Dense gases are single-phase fluids with a molecular complexity such that the fundamental derivative of gas dynamics [START_REF] Thompson | A fundamental derivative in gasdynamics[END_REF]

Γ := 1 + ρ c ∂ c ∂ ρ s
(where ρ is the density, p the pressure, s the entropy, and c the sound speed), which measures the rate of change of the sound speed in isentropic transformations, is less than one in a range of thermodynamic conditions close to the saturation curve. In such conditions, the speed of sound increases in isentropic expansions and decreases in isentropic compressions, unlike the case of perfect gases. For dense gases, the perfect gas model is no longer valid, and more complex equations of state must be used to account for their peculiar thermodynamic behavior. Moreover, in the dense gas regime, the dynamic viscosity µ and the thermal conductivity λ depend on temperature and pressure through complex relationships. Similarly, the approximation of nearly constant Prandtl number Pr=µc p /λ is no longer valid. Numerical simulations of turbulent dense gas flows of engineering interest are based on the (Reynolds-Averaged Navier-Stokes) RANS equations, which need to be supplemented by a model for the Reynolds stress tensor and turbulent heat flux. The accuracy of RANS models for dense-gas flows has not been properly assessed up to date, due to the lack of both experimental and numerical reference data. DNS databases [START_REF] Sciacovelli | Dense gas effects in inviscid homogeneous isotropic turbulence[END_REF][START_REF] Sciacovelli | Small-scale dynamics of dense gas compressible homogeneous isotropic turbulence[END_REF] are then needed to quantify the deficiencies of existing turbulence models and to develop and calibrate improved ones. In this work we first summarize some recent direct numerical simulation (DNS) results [START_REF] Sciacovelli | DNS of supersonic turbulent channel flows of dense gases[END_REF] for supersonic turbulent channel flows (TCF) of PP11, a heavy fluorocarbon representative of dense gases, at various bulk Mach and Reynolds numbers. The most relevant effects are represented by non-conventional variations of the fluctuating thermodynamic quantities, compared to perfect gases and a strong decoupling between thermal and dynamic effects almost everywhere in the flow, except in the immediate vicinity of the solid wall. Preliminary considerations about the validity of some currently-used models for the turbulent stresses and heat flux are carried out based on a priori comparisons between the exact terms computed from the DNS and their modeled counterparts.

Governing Equations and Numerical Method

Dense gas flows are governed by the compressible Navier-Stokes equations, supplemented by suitable thermodynamic and transport-property models. In the present work, the gas behavior is modeled through the Martin-Hou (MAH) thermal equation of state [START_REF] Martin | Development of an equation of state for gases[END_REF], which is reasonably accurate for the fluid of interest and requires a minimum amount of experimental information for setting the gas-dependent coefficients. In addition thermodynamic models relating the dynamic viscosity µ and thermal conductivity κ to the gas temperature and pressure have to be specified. In the present calculations, the transport properties follow the Chung et al. model [START_REF] Chung | Applications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity[END_REF], which incorporates a correction term in the dense-gas region. The working fluid is perfluoro-perhydrophenanthrene (chemical formula C 14 F 24 ), called hereafter with its commercial name PP11, often used in dense-gas studies. DNS were also carried for air, modeled as a polytropic perfect gas. In the latter case, the viscosity is assumed to follow a power law of the temperature and the thermal conductivity is computed according to a constant Prandtl number assumption. The governing equations are approximated in space by means of optimized finite difference schemes, supplemented by an optimized selective sixth-order filter. A low-storage six-step optimized Runge-Kutta is used for time integration.

DNS results

A parametric study was carried out at three bulk Reynolds numbers Re B :=

ρ B u B h µ w
(3000, 7000 and 12000) and three bulk Mach numbers M B := u B c w (1.5, 2.25 and 3.0) [START_REF] Sciacovelli | DNS of supersonic turbulent channel flows of dense gases[END_REF]. Details about the computational setup and grid resolution can be found in [START_REF] Sciacovelli | DNS of supersonic turbulent channel flows of dense gases[END_REF]. In the following, the subscripts (•) B , (•) w and (•) CL denote time and space averaged values over the channel cross-section, at the wall and at the centerline, respectively; the semi-local friction velocity. This mixed scaling provided quite satisfactory results in collapsing first-and second-order moments for a wide range of M B . For PP11, due to the large specific heat of the fluid, the average temperature is almost constant across the channel for any choice of the Mach and Reynolds numbers, and the centerline temperature differs less than 1% from T w . Decoupling of dynamic and thermal effects in the dense gas also leads to smaller mean density variations across the channel. The normalized viscosity µ/µ w , which follows temperature variations for air, varies instead like the density for PP11 and tends to decrease toward the channel center. As a consequence the friction Reynolds number Re * τ increases toward the channel centerline in PP11. Specifically, the dense gas flow exhibits lower values of Re * τ near the wall, compared to a perfect gas flow at the same bulk conditions, whereas the centerline value of Re * τ is much higher in the dense gas, due to the negligible friction heating in the outer region. On the other hand, the average Prandtl number Pr decreases from the wall to the centerline, following essentially the same trend as the average specific heat. Figure 1 displays the distributions of the above-mentioned quantities across the channel, as a function of y * . Dense gas effects are stronger at higher M B , since the local thermodynamic states spread over a wider range. Sample results for second-order statistics are reported in figure 2, which displays the r.m.s. values of the density, the Reynolds shear stress in semilocal scaling ρu ′′ v ′′ + = τ -1 w ρu ′′ v ′′ , as well as the ratio of the turbulent kinetic energy production to dissipation. The relative density and pressure fluctuations are of the same order of those observed for air flows (see [START_REF] Sciacovelli | DNS of supersonic turbulent channel flows of dense gases[END_REF]), whereas temperature fluctuations (not reported) are nearly two orders of magnitude lower. Remarkably, ρ ′2 decreases monotonically from wall to centerline, contrary to light gases. This is due to the peculiar thermodynamic behavior of PP11 at the considered conditions, as demonstrated in [START_REF] Sciacovelli | DNS of supersonic turbulent channel flows of dense gases[END_REF] by using the equation of state. In all cases, density fluctuations remain small compared to the mean value, and Morkovin's hypotesis is satisfied even at the highest Mach number. Despite the striking differences in the thermodynamic behavior, Reynolds stress profiles are similar to those observed, e.g., in [START_REF] Patel | Semi-local scaling and turbulence modulation in variable property turbulent channel flows[END_REF] for low-Mach TCF with temperature-dependent transport properties. The liquid-like behavior of viscosity leads to an increase of the spanwise, wall-normal and Reynolds shear stresses with respect to the corresponding incompressible evolution, whereas the streamwise one decreases. This effect is stronger when increasing Mach number. The last subfigure shows the ratio of production to dissipation term P k /ε k of the turbulent kinetic energy budget. The production peak is located as usual at y * ≈ 12. For the higher Reynolds number, a second production peak is observed in the outer region, like in high-Re incompressible flow, due to the reduced dissipation close to centerline. A visualization of flow structures for M B = 3, Re B = 12000 is provided in figure 3.

A priori analysis of turbulence models

DNS data are used to investigate the validity of some popular models for the RANS equations. More specifically, we focus hereafter on the eddy-viscosity and turbulent Prandtl number assumptions, commonly used to approximate, respectively, the Reynolds stress tensor and the turbulent heat flux. To this end, "exact" eddy viscosity and turbulent Prandtl number are computed from DNS data as: where u is the average streamwise velocity and λ t = ρv ′′ h ′′ dh dy -1

µ t = ρu ′′ v ′′ du dy -1 ; Pr t = c p µ t /λ t (1)
-• -• -• • • • • • • • • • ---- -• • -• • -
is a turbulent thermal conductivity, with h the average static enthalpy. Input quantities required by eddy models are also based on DNS. In this study we restrict our attention to two low-Reynolds variants of the kε model, namely, the Launder-Sharma (LS) [START_REF] Launder | Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc[END_REF] and Chien (CH) [START_REF] Chien | Predictions of channel and boundary-layer flow with a low-Reynolds turbulence model[END_REF] models, which assume that µ t = C µ f µ ρk 2 /ε, with C µ usually taken equal to 0.09, the damping function f µ is exp(-3.4/(1 + R t /50) 2 ) (R t = ρk 2 /(µε)) for LS and 1 -exp(-0.0115y + ) for CH. Durbin [START_REF] Durbin | Near-Wall turbulence closure modeling without "damping functions[END_REF] showed that a more appropriate choice for the velocity scale in the inner region is represented by the root-mean square of the wall-normal fluctuating velocity, and reformulated the eddy viscosity as µ t = C v ′′2 k/ε with C = 0.2. In figure 4 we report the results for the highest Reynolds number, both for air and PP11. In both cases the LS model badly overestimates the turbulent viscosity, especially in the viscous sublayer, as also observed by other authors (e.g. [START_REF] Durbin | Near-Wall turbulence closure modeling without "damping functions[END_REF]) for incompressible flows. Chien's model captures better the trend of µ t but also overestimates the turbulent viscosity in the outer region. Durbin's model provides a reasonably accurate approximation of the µ t profile up to part of the logarithmic region. For air, the model departs from the DNS profile at y * ≈ 40 and, unlike the two preceding models, it underestimates µ t in the outer region. For PP11, the model remains in rather good agreement with DNS up to y * ≈ 150 and overestimates µ t for higher values of y * .

In figure 5 we report the exact turbulent Prandtl number at various M B and Re B . For air, Pr t follows the trends observed by other authors in the literature (e.g. [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF]). In particular, for the present relatively low-Re flow, Pr t exhibits only a small plateau around y * ≈ 100 where its value is close to the standard "constant" value of 0.9. In the outer region, Pr t decreases with y * , while in the inner region it exhibits a local maximum at about y * = 50 and tends to approximately 1.1 at the wall. For PP11, the overall behavior is rather close to that of the perfect gas over most of the channel height (y * 5). However, the solution exhibits a larger '0.9' plateau than the air flow, located at y * ≈ 120, most likely because of the higher local Re * τ . The local maximum around y * ≈ 50 is also observed for the dense gas. A drastically different behavior is observed in the viscous sublayer (y * 5), where the local Prandtl number is much higher and the local Re * τ much lower than in air. In this region (y * ≈ 2), Pr t exhibit another local maximum (more or less pronounced according to the flow conditions) and tends to values lower than 1 at the wall (≈ 0.4 at the highest Mach number). Inspection of the DNS data for µ t and λ t suggests the following explanation: for air, both µ t and λ t decrease approximately at the same rate (O(y * 3 )) when y * → 0. Since in this case c p = const, then Pr t tends toward a constant nonzero value. For PP11, λ t decreases at a lower rate than µ t in the near wall region, due to the smaller enthalpy variation in this high-c p fluid. As a consequence, their ratio tends to vanish at the wall. However, c p increases abruptly when y * → 0 (following an approximately exponential trend), hence the local maximum. Direct numerical simulations of plane turbulent channel flows of dense gases have been performed at various bulk Mach and Reynolds numbers. For a dense-gas, the classical y + scaling based on the friction velocity fails to collapse thermodynamic profiles and Reynolds stresses at high M B , and semi-local scaling, which accounts for variations of the flow properties, has to be adopted instead. Due to the high specific heat, coupling between dynamic and thermal effects is found to be very small for the dense fluid. Turbulence structure is shown to be little affected by dense gas effects. For the adopted thermodynamic conditions, transport properties exhibit a liquid-like behavior and the local Reynolds number in the outer region is found to be much higher than in corresponding air flows. A priori analyses of the validity of some common modeling assumptions for the eddy viscosity and turbulent Prandtl number showed that, for a dense gas, turbulence models for eddy viscosity follow the exact trend more closely than in perfect gas (at the present high Mach numbers), due to the higher local Reynolds number, but eddy viscosity is overestimated. An adjustment of the model constant could help improving the results. A peculiar behavior is observed for the turbulent Prandtl number close to the wall, which peaks more or less abruptely in the viscous sublayer, instead of tending to a constant value. Further investigations of the near wall thermal behavior are planned as future research.

  (•) indicates Reynolds averaging and (•) ′ Reynolds fluctuations; similarly, (•) and (•) ′′ denote Favre averages and fluctuations. Results for the different cases were compared by introducing an empirical semi-local scaling initially proposed by Huang et al. [7] for compressible flows, which corrects the usual wall scaling with centerline quantities. Specifically, y * = ρ(y)u * τ y µ(y) and Re * τ = Re τ ρ(y) ρ w µ w µ(y) , with u * τ := τ w ρ(y)

Fig. 1 Fig. 2

 12 Fig. 1 From left to right: local friction Reynolds number (Re * τ ), average Prandtl number (Pr), and average isobaric specific heat normalized with the gas constant (c p /R) as a function of y * for DNS of PP11 TCF at various M B and Re B .

Fig. 3

 3 Fig. 3 Isosurface of Q(h/u B ) 2 = 1 coloured with streamwise velocity (M B =3, Re B =7000).

Fig. 4 Fig. 5

 45 Fig. 4 Exact and modeled eddy viscosity (ν t /ν w ) for air (left) and PP11 (right) at M B = 3 and Re B = 12000. ---: LS; ---: CH; -• -• -: Durbin; : DNS.

Acknowledgements This work was granted access to the HPC resources of GENCI (Grand Equipement National de Calcul Intensif) under the allocation 7332.