Interdisciplinary Group for the Study of the School Standards [ http://grip.ujf-grenoble.fr/ ]
Groupe de
Réflexion Interdisciplinaire sur les
Programmes
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A Historical Testimony for the French Primary School in the Context of Concrete vs. Abstract Numbers Michel Delord Numeracy and Beyond Banff, December 4 - 9, 2004 [ http://www.pims.math.ca/birs/workshops/2004/04w5044/index.html ]
A Historical Testimony for the French Primary School in the Context of Concrete vs. Abstract Numbers Michel Delord - Banff, December 4, 2004 [ http://michel.delord.free.fr/grip-banff -eng.pdf]
Though it has been asked many times, there is one question I shall not dwell on, since I have so often anwered it in articles, books and talks; it is this: are sciences more important to humanity than letters, and should one therefore give a scientific rather than a literary education to children; or should one give them a literary rather than a scientific education? Here is what I invariably answer: you might as well ask whether it is more important for people to eat or to sleep; whether it is preferable to deprive them of food and allow them to sleep, or to deprive them of sleep and allow them to eat. I declare that in one case as in the other, things will finally happen in exactly the same manner, and the result will be, in a very short time, the passage from Life to the Beyond, for any person kept on such a diet. But, for a long time now, we have been doing more or less that very nonsense with the two halves of French youth, the literary category and the scientific one. By practising literary as opposed to scientific education, by training lawyers who have no idea of how a locomotive works, alongside whom one could imagine engineers with perhaps very strong mathematical backgrounds but unaware throughout their lives, that there was ever a man called Rabelais and another named Paul-Louis Courier, we shall create two castes of half-men, but never a humanity, nor a society, nor a fatherland. It is even shameful and humiliating, among people who call themselves civilised, to think that such a question could ever have been aked! Charles-Ange Laisant, L'éducation fondée sur la Science, F. Alcan (1904), p. 71 Since the counter-reform of 1970, there has certainly been progress in this direction. The elimination of teaching concrete numbers has moreover meant the separation of mathematics from its "areas of application": to go with the literati without any notion of science, school now also produces students of mathematics who don't know physics and physicists who don't know mathematics. It has thus passed from the hand-crafted production of half-men to the industrial output of fractional people, quarters, eighths, and so on ... which thus constitute the elite of our nation. I shall try, beginning with examples from what school was around 1900, to show glimpses of its greatness, which might help us modestly to find the way back to its unity. Michel Delord, November 23, 2004 *** A brief comment from Rudolf Bkouche (In French, sorry) : Quelques remarques sur le texte de Banff http://michel.delord.free.fr/banff-rb.pdf *** - "The programmes of the French primary school are, on average, one or two years ahead of those of other countries." - The problematic of the counter-reform of the Seventies - The intuitive method: The example of reading and calculating in Grade 1 - The main point of rupture in 1970: The elimination of operations on magnitudes - The transition to abstraction and logic - Two objectives regarding calculations with magnitudes i) Introduction to dimensional analysis: No orders of magnitude without magnitudes Higher mathematics and elementary arithmetic ii) Numerical calculation: from arithmetic to field axioms. Appendix: - Brief history of French national arithmetic demands for grade 1 and 5: A slow but sure and massive comeback to Middle Ages' pedagogy - A 1956 Grade 1 exercise book (also at: http://michel.delord.free.fr/cp56p.pdf )
***
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On the History of French Primary Schools Logic is for mathematics what the skeleton is for an animal, which would not be able to stand up without it; but it would be a strange zoology which never dealt with anything but skeletons. Felix Kleini
"The programmes of the French primary school are, on average, one or two years ahead of those of other countries." The French primary school was, for a long time, one of the best in the world. This was attested as early as the 1880's, when Friedrich Engels wrote in a letter, dated October 28, 1885, to August Bebel: "At this time, the French have the best schools in the world". The phrase "at this time " is a direct reference to the laws put in place from 1881/1882 onward by Jules Ferry, Minister of Public Instruction. He imposed in particular what interests us here: mandatory national curricula for primary school, which actually varied very little from that time until 1970, that is, for almost a century. This high level of the French primary school was also attested, in a contrary spirit, by Antoine Prost, one of the main theoretical artisans of the dumbing down reform of the 70's, which was to take the philosophically opposite view of the preceding programmes and progressions. Here is an excerpt from his book describing the school of the 50'sii :
"One finds, for instance, in the curriculum for the Grade 5th, operations with fractions and grammatical niceties about the past participle which 75% of the students will master only at age 13-14. According to international comparisons made by R. Drottens in 1954, French children learn to conjugate verbs two years earlier than their German or Dutch counterparts ; they begin logical analysis [of sentences, K.H.] two years before the Germans and four years before the Italians; they must know how to count to 1000 while the most advanced of their neighbours stop at 20 ; they learn multiplication and division by two-digit numbers one year before the Germans and the Dutch, two years before the Belgians and the Italians. While the Belgians and the Dutch begin to work with percentages in their 5th year of school, and the others in their 6th, the French struggle with them from their 4th year on. Truly, one of the characteristics of French education is exactly to Division is a difficult art, especially when you are so force-feed notions to children too young to little. French primary curricula are, on average, one or assimilate them, or to demand of them behaviour two years ahead of those of foreign schools, but almost which they cannot yet perform physically." half our students have to repeat at least one year.
If, on average and for almost a century, French primary school was two years ahead of schools in the other countries mentioned, it is not without interest to understand some of the reasons for its success -- and that is what I wish to pursue here.
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The problematic of the counter-reform of the Seventies That reform was a complete reversal of the preceding notions of school. It corresponds to putting in place a general pedagogy centered on the reform of teaching Mathematics - "the new Math"- and of French iii for which the introduction of the communicative approach would eventually replace the "scholarly" study of language, which meant to go to school…to learn what we know of language without going to school (Henri Poincaré iv ). It is important to note the historical role of the new didactics, invented at this time as the pedagogy of mathematics; it introduced for mathematics concepts like the didactic transposition 1 which later spread to all subjects. Two factors which converged to increase the cost of schooling, would have a fundamental influence on the decision makers: - The desire to prolong the duration of compulsory schooling, from 14 years of age since 1938 to 16 years in 1959 (reform of Berthoin whose complete effect did not exist until 1967) - The post-war Baby Boom required a greater number of students to be sent to school Faced with this double threat of an increasing school population, the political and economical decision makers2 , panicked by the spectre of a "student explosion"3 would be responsive to any form of argument which allowed costscutting by reduction. Among these two were fundamental: - i) Teaching abstraction from the start. For example, - Immediately doing "pure mathematics" - i.e. teaching directly axiomatic and structures - in primary school, instead of arithmetic and calculation: French reformers, in their first manifesto (Charte de Chambéryv, January 1968), claim that they want to teach to all school's levels, "Modern Mathematic, better called constructive, axiomatic and structural conception of mathematics" - Or imposing on high school part of the teaching normally reserved for the university: "This meeting, the Royaumont Seminar, took place in the autumn of 1959 in France. Together with an associated survey of current practice, it had been conceived within the OEEC earlier in 1959 for "the purpose of improving mathematics education" for "university-capable" pupils.. Since there is no turning back, nor hope of lengthening the years of study devoted to mathematics, there is a "squeeze" in the course of this study. The only solution is for the secondary school to take on some of the burden now resting on the university [underlined by me, MD], perhaps as much as is compatible with the intellectual ability of secondary school pupils"4
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This concept, false dichotomy between scholarly knowledge and teachable knowledge, comes from the impossibility to teach new maths in primary schools. It is one of the innumerable false dichotomies which produces i) real difficulties for teaching in primary schools ii) new false dichotomies et more difficulties to teach in upper levels Cf. Rudolf Bkouche, De la transposition didactique, http://michel.delord.free.fr/rb/rb-transpo.pdf 2
From this time on, there is a permanent bond of cooperation between "pedagogical research" and the decision makers. Since the latter also pay the bills, the only well-funded research is the kind whose results permit local and short-term savings in operating costs, even if its real effects are catastrophic. A particuliarly flagrant example of this is barring the possibility of a grade being repeated. 3
The title of Louis Cros’s 1961 bestseller. Quoted as a reference in the New Math manifesto Charte de Chambéry. http://michel.delord.free.fr/chambery.pdf 4
Renegotiating secondary school mathematics: A study of curriculum. Change and stability, Barry Cooper, The Falmer Press, 1985. http://membres.lycos.fr/sauvezlesmaths/Textes/IVoltaire/cooper.htm
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- ii) Making the programs lighter. Since, according to Antoine Prost, the frequency of redoing a year in the French school system was due to its teaching subject matter to students too soon, a pedagogy was introduced which lightens the content under the general pretext that a student learns better if there is less to learn . This affirmation is certainly true for a teaching reduced to learning procedures and skills of immediate use, allegedly practical but in fact blind, exclusively focused on memorization where one does not understand what one learns. But - it becomes completely false as soon as one aims at the learning of notions which are logically interrelated because, in this case, the elimination of one link in the chain of reasoning makes the learning of the remaining notions more difficult or even impossible: " Insights are fertile only when they are intertwined. If we get attached only to those which might yield immediate results the intermediate links will be missing and we will no longer have a context.."(Henri Poincaré vi ). - it creates by itself a dynamics of progressive thinning out : “an infernal spiral which pretends to facilitate comprehension by thinning out fundamental knowledge. The result is exactly the opposite: ‘Swiss cheese structure’ of curricula makes the understanding of fundamental know-how more difficult if not impossible. This will serve as a pretext for further alleviation, but above all it destroys in the child any possible access to rationality. Instead, it systematically teaches him to ‘think’ incoherently and reduces learning to procedural contents which even can no longer be mastered because the very mastery of their mechanisms presupposes a minimum of rationality."(Petition against new primary school curriculum, November 2001 vii) * * * In this sense, the continuing problematic effect of the counter-reform of the 70’s - i) is not only an alleviation but a modern form of obscurentism, - obscurentistic and medieval in the historical sense as is shown below by examples of Grade 1 curricula which in effect return to teaching methods from before the age of Enlightenment. - modern because it sees no sense in the contents taught except for their applications (obsession with manipulatives5 ), partly because it includes the utilitarian aim for getting a patent 6 , which was certainly not the case in historical scholasticism. ii) is worse than the formal conjunction of historical scholasticism and empiricism: - Analysed historically these are not regressions because the dynamics of the evolution of scholasticism constitutes progress, and experimentation itself is a means of overcoming the dogmatism of revealed truth. - Historical scholasticism safeguarded the “logic of contents” which the dynamics of the reforms has destroyed by emptying the notion of curriculum of its content when emptying the curricula themselves of their contents. Roger Bacon certainly did not aim at getting a patent. iii) makes, precisely because it is not a simple alleviation, the attendance of school and the increase of hours, for the first time since its the foundation in the 19 th century, not a sign of progress but even permits a mental regression in students. This fundamental point, which partially justifies the success of homeschooling will not be developed here at length: without mentioning the afflux of patients to orthophonic clinicsviii as a result of modern reading instruction, one could briefly touch on the example of mental arithmetic which would be better not taught at all than taught the way it is. In fact, ever since it has been taught as “calcul réfléchi”, which mixes written and mental arithmetic, actually teaching mental arithmetic means investing a lot of time to unlearn reflexes formerly taught. 5
La main à la pâte, the French equivalent of Hands on
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The aim of learning science, against mathematics on chalkboard, is to get patents according to two important present decision makers of French Ministère of Education. See: Philippe Joutard et Claude Thélot, Réussir l'école, Pour une politique éducative, Le Seuil, 1999, 292pages, page 179.
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The intuitive method: The example of reading and calculating in Grade 1 In contrast to the theories of alleviation, the effectiveness of teaching in the years 1880 to 1970 is based, to accelerate learn ing, on the synergy brought about by learning different subjects and the various concepts related to one another within the same subject : whence the importance of the coherence and compactness of curricula (then called plans of study). This design is one of the bases of the "intuitive method", a pedagogical method, intended first and foremost for elementary teaching, whose principles were repeated at the beginning of every Instructions Officielles until 1945. As a characteristic example one could take the curriculum of Grade 1 as explained by its principal theoretician, Ferdinand Buisson (1841-1932). He was the director of elementary teaching at the Ministry Public Instruction between 1872 and 1896 and the auteur of the monumental Dictionnaire de pédagogie et d'instruction publique7 (7000 pages in 4 volumes) intended as a reference work for teachers and written by the intellectual elite of the time ( For example, Viollet-le-Duc for architecture) . He also received the Nobel Peace Prize in 1927. He describes it as follows. . What exactly constitutes the intuitive method in those elementary studies which cannot be limited to object lessons ? In a certain progression of teaching which leaves the child the pleasure and benefit, if not of discovery and surprise, that would be promising too much, then at least some initiative and intellectual activity. Even if one does not show objects or images, one can be said to be teaching by intuition every time one chooses, instead of making him follow the teacher passively and repeat a ready-made lesson with docility, to provoke him to search, to help him find, to put him on the path (as an old and very apt image would suggest) and then leave him the merit of taking a few steps by himself. One could almost say that there are two kinds of logic: that of the child and that of the adult, one entirely natural and intuitive, the other more informed., more considered, more methodical. It is a great temptation for the teacher to follow the latter, which is the only rational one, the only one which satisfies his own mind, his need for connectedness and regular deduction: it is the one which is really natural for the mature person. It proceeds from simple to complex, from principle to consequence, from rule to example. And that`s precisely what tires and disheartens the child. And the older methods were inexorable, in the name of logic, on the necessity of its interminable preliminaries. Was the child to learn how to read? Then one had to begin by teaching him all the letters and thereafter their combinations in syllables, before arriving at a word, let alone a sentence. What a desert to traverse for the poor little intelligence! From reading one went on to writing, proceeding in the same manner: not first the word or even the letter, but downstroke and vertical lines. Who does not remember the long pages of vertical lines of his first school? And likewise when some other subject was approached: in geography, nomenclature and definitions of all geographical terms learnt by heart, and then the definition of the earth, its division into oceans and continents, their enumeration and the enumeration of their subdivisions, all of it before arriving at a single name familiar to the child, or at a single known object. Was all of that absurd, illogical, unreasonable? Not at all. It was the progression of a mature mind, which, knowng how to resolve the subject to be studied into abstract ideas, first takes the simplest ones and gradually concatenates them into increasingly complex combinations rigorously subordinated to one another. How different is the movement of a child's mind, which quickly and joyously wants to run from known to unknown, from concrete to abstract, from easy to difficult, by leaps rather than step by step. Sometimes people say that the child's intelligence is capricious: it is not; it only seems so to us because it does not have the continuity and regularity of ours; it loves to guess, to discover, to enjoy learning instead of buckling down to it, to enjoy above all the sense of its power and freedom, to feel itself working. For exercises of the mind, children show the same dispositions as for those of the body: regular and monotonous long walks drain and exasperate them, a methodical exercise in gymnastics yields no recreation unless it is very short; but let them run freely, frolic as they please, change exercises and exercise without knowing it, then they are indefatigable. The intuitive method, as it is applied today in every subject of elementary 7
See some excerpts at : http://michel.delord.free.fr/dp.html
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teaching, has no other aim than to take into account this need for spontaneity, variety, and intellectual initiative on the part of the child. For reading, instead of having to plow through all the letters and all those syllables devoid of meaning, he is given, as soon as he knows two or three letters, little words which occupy his mind, satisfy his imagination, and stimulate his curiosity for the following lessons, each lesson thus carrying, as it were, its own reward: the logical order may suffer by this, and the child must more than once supply, by a sort of divination or intuition, what he lacks strictly speaking to be in a position to decipher a word, but that is precisely where he finds his pleasure; the obstacle has been overcome, and he is filled with the pride of a conquest just made; he is not yet at the age to feel the need for keeping track meticulously and consciously of the procedures followed, he just wants to keep going. Later on, there will be time to have him analyse what he has now grasp ed with a true but too rapid glance. In geograpphy, his interest is at first engaged in near-by things right before his eyes: and by analogy he is led to comprehend, progressively extending his horizon, all the great phenomena he has not seen, with the help of the small ones he does see.. In arithmetic, he is not immediately shown abstract numbers with their relations and laws: it is on concrete objects that his attention is first exercised, and his senses are used not to make him rely on them throughout his life, but to teach him to do without them: the day is not far off when he will be able to perform, in his head and by intuition, operations whose rigorous justification will not be within his reach until many years later. There is no child who cannot mentally and effortlessly do subtractions, multiplications, divisions, and indeed fractions, involving the first ten nmbers, long before suspecting even the names of the four operations. In grammar, and there perhaps more usefully than anywhere else, the child's intelligence can be entrusted to itself, challenged to find the rule and not always limited to applying it passively, encouraged to proceed by analogy, to produce on its own the generalisations which the textbook gives no doubt ready made and classified, a free effort of the mind, a true exercise of thought and word. [Ferdinand Buisson, Article Intuition et méthode intuitiveix, in Dictionnaire de pédagogie et d'instruction primaire, Hachette, 1887. Tome 2 de la première partie, pages 1374 à 1377.] Its great strength, as compared with the preceding methods, which F. Buisson justly describes as archaic, scholastic and medieval, comes from relying on : - i) the simultaneous learning of reading and writing (a method called writing-reading) whereas the preceding methods separated the learning of these. -ii) the simultaneous learning of counting and calculating or more precisely the simultaneous learning of the 4 operations as counting progresses ( Intuitive calculation8 ) whereas the precedin g methods first taught counting and then successively each operation separately: in fact, - counting is tied in with the operations : 340 surely signifies 3 times 100 plus 4 times 10 - each operation is defined in relation to the others - the "intimate knowledge of number" (René Thom) does not come about unless a number is conceived as the result of various operations : one does not really know 6, once its place in the counting sequence (between 5 and 7) is established, except by knowing the results of 4+2, 5+1, 7-1, 8-2, 2×3, 6×1, the quotient by 2 of 12 and 13 … But let us hear what F. Buisson has to say : " Apart from the psychological considerations which inspired it, [the method of intuitive calculation] lets children, on their own and by intuition, perform the essential operations of elementary computation; its aim is to make them familiar with numbers: being familiar with an object means not just knowing its name, it means having seen in all its forms, in all its states, in its diverse relations with other objects; it means being able to compare it with others, to follow it in its transformations, to grasp and measure it, to compose and decompose it at will. Thus treating numbers like any other objects to be 8
See Ferdinand Buisson, Article Calcul Intuitif http://michel.delord.free.fr/fb-calcintuit.pdf
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presented to the child's intelligence, Grube strongly opposes the old custom of successively teaching the pupils first addition, then substraction, then the other two operations." [F. Buisson, Intuitive Calculation] - iii) a subtle understanding, regarding mental arithmetic, of the initial linkage between the learning of speaking and reckoning 9 . In fact, mental arithmetic appears in the Dictionnaire pédagogique with two complementary objectives : - a) a general one based on the fact that mental arithmetic side-steps written enumeration: the student does not have to imagine "the language of numbers as being necessarily a written one. On the contrary, it would be essential to familiarise him with the idea that spoken arithmetic precedes written arithmetic", both to counteract "the all too common custom of writing even the simplest computations", and also to cultivate "intellectual gymnastics of the greatest importance, leading to habits of analysis and reflection which very quickly increase the perspicacity of the mind", for "it is by this that the mind, in some sense, assimilates the essence of its arithmetical lessons, and harvests all their fruit". - b) in Grade 1, as intuitive computation which provides "a way of teaching the first elements of reckoning", and as a form of "purely oral" computation preceding written calculation for small numbers. This introduction of intuitive computation aims of course at preparing the use of mental arithmetic throughout all schooling. It permits : - actually teaching the use and habit of mental arithmetic instead of just proclaiming its merits, and teaching really mental techniques (based on non-written enumeration) since the pupil has no other modes of compution at this stage, - at the same time showing the necessity and superiority of written calculation, which applies beyond the reach of mental arithmetic. Moreover, as is easily shown by the following three examples, the modernity represented by the principal discoveries of pedagogy in the last forty years is a return to scholastic me thods which rely excessively on memory and rote to the detriment of intelligence and intuition : - i) functional or balanced methods once again separate writing from reading as pupils systematically "read" items they do not know how to write - ii) there has been a progressive return to the separation of learning how to count and how to compute (cf. in appendices Brief history of French national arithmetic demands for grade 1 and 5) - iii) the introduction of "reasoned computation", which blurs the boundary between mental and written calculation, has destroyed what constitutes the strength of mental arithmetic: its character as an entirely unwritten form of computation.
The main point of rupture in 1970 : the elimination of magnitudes This is not based on a personal analysis made a posteriori, but on a declaration published in 1972, by the APMEP 10, the principal association of teachers of mathematics and the hub of action for the "New Math" reform, in their special issue dedicated to the Bulletin Officiel of January 1970x, which had introduced this New Math in elementary school. It is therefore a central and definitive statement: The elimination of "operations with magnitudes " is the truly fundamental mutation brought in by the transititional curricula, the one which profoundly transforms the thought processes in elementary teaching.xi This elimination of operations with magnitudes -- and hence with concrete numbers -- is not argued on its merits by the B. O., but appears in the following form: Sentences like 8 apples + 7 apples = 15 apples are not part of the language of mathematicsxii 9
See last chapter : Arithmetic and language
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Association des Professeurs de Mathématiques de l'Enseignement Public
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This is of course a pedagogical absurdity, but above all a mathematical one, since in 1968, i.e., two years before the publication of the B.O. and four years prior to the commentary by the APMEP, the great geometer Hassler Whitney had published an article providing an axiomatic, "modern" mathematical framework for operations with magnitudes, entitled The Mathematics of Physical Quantities11 . In it, he explicitly says -- and shows by giving an underlying mathematical structure (rays and birays) -- that it is entirely "mathematical" to write : 5 cakes + 2 cakes = (5+2) cakes = 7 cakes or, for instance,
2 yd = 2 ( 3 ft) = 6 ft
The context of his Introduction makes it clear that he is explicitly taking aim at the New Math, notably by pointing out the absurdity of imposing stilted language like "Complete: 2 cm measure the same as … mm; 80 mm measure the same as … c m." xiii, when he says: The fact that "2 yd" and "6 ft" name the same element of the model enables us to say they are equal; there is no need for such mysterious phrases as "2 yd measures the same as 6 ft." This proves among other things that the practice of calculating with magnitudes is rather more "modern" than the reduction of all calculation to that on pure numbers. And thirty years after 1970, we are basically still at the same point, as one can read in the official document Accompagnement des programmes de troisième de collège, i.e. Curriculum guide for Grade 9: In fact, in mathematics there is no such thing as working with magnitudes (that is the concern of other disciplines, such as physics, technology, the life sciences, earth sciences, or geography and economics, for example).xiv
The transition to abstraction and logic . Ferdinand Buisson 1882
W. P. Thurston 1994
The primitive tendenccy in pedagogy … is that of all teachers at the beginning of their careers : start with the general idea of the science to be taught, decompose it logically into a certain number of abstract notions, define each of these notions, make the students learn these definitions, then deduce rules or formulas from them, and continue this way, constructing definition after definition, chapter by chapter, the whole theoretical edifice of the science, except for making them then do applications in the forme of exercises, problems, and examples.
In caricature, the popular model holds that D. mathematicians start from a few basic mathematical structures and a collection of axioms "given" about these structures, that T. there are various important questions to be answered about these structures that can be stated as formal mathematical propositions, and P. the task of the mathematician is to seek a deductive pathway from the axioms to the propositions or to their denials. We might call this the definition-theorem -proof (DTP) model of mathematics.
Ferdinand Buisson, Dictionnaire Pédagogique, Article Abstraction
William P. Thurston (Fields medal 1982) On proof and progress in mathematics
Once it is admitted that the teaching of mathematics aims in part at the students' mastery of logically related abstractions, something which, incidentally, is not limited to mathematics, since all rational thought is abstract, the question of the passage to abstraction does indeed arise. The first error consists in skirting the issue by ... directly teaching abstraction. The other one is to imagine an absolute opposition a priori between THE abstract and THE concrete : without denying the existence of an antithesis, one could, in the context of learning (as in the history of 11
The Mathematics of Physical Quantities Part I: Mathematical Models for Measurement, February 1968 Part II: Quantity Structures and Dimensional Analysis, July 1968 In American Mathematical Monthly. Vol. 75. The introduction can be found at http://michel.delord.free.fr/h_whitney.pdf
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ideas), consider abstraction as an action that "extends a truth by removing elements which make it particular" (D'Alembert and Diderot), and leads to an ongoing process in which each following stage is somehow an abstraction of the preceding more concrete one. Ferdinand Gonseth's book "Les mathématiques et la réalité"12 (1935), has an excellent description of this for both the genesis of the notion of number and that of point and line. We shall here concentrate on the notion of whole number. To give some examples (which are simplistic compared to Gonseth's much wider perspective), it could be pointed out that the notion of the whole number 10 is more abstract than the notion of 10 apples or that of 10 kilometers. Nevertheless, 10 apples already represents a level of abstraction because, thinking of 10 apples presupposes, since 10 strictly identical apples do not exist, being able to leave aside colours and sizes (and wanting to count). But thinking 10 kilometers is a little more abstract than 10 apples, since it refers to a distance which cannot be directly perceived by the senses and whose comprehension supposes, if not that of the metric system (which is a system, i.e., a theoretical concept), at least that of the connection between the kilometer and the meter (which recovers the lost distinction between ficticious units and effective units). Ferdinand Gonseth explains: "In the evolution of the concept of whole number, at leat three rather different periods can be distiguished: the first one precedes any conscious attempt at systematisation; the second - specifically arithmetical - one is characterised by an explicit formulation of the theory of whole numbers; only the third one reaches a level involving "logic". The passage from one period to the next is accompanied by a deep transformation of the very essence of number. It is interesting to consider the passage from the arithmetical to the logical stage in greater detail because it will lead us to define the notion of a unit and hence that of a concrete number, that is, a number followed by the name of a unit. To do this - consider the set of natural numbers N : {1, 2, 3, 4, 5, 6,…n…} and the set of multiples of 2 , denoted by 2N: {2, 4, 6, 8, 10, 12 …2×n…}. - endow the set 2N with two laws of composition - one of them denoted + : (2×n) + (2×m) = 2×(n+m) - the other denoted * : (2×n) * (2×m) = 2×(n×m) The structure (2N, + , *) is certainly isomorphic to (N, +,×) via f(n) = 2×n.. Hence 2N is certainly a model of N. And this logical - axiomatic structure is what N has in commn with 2N (but also with 3N, 4N, 5N,….) This logical aspect could be presented in the form of the following axioms, which are certainly satified by (N, + , ×) and all the (nN, + , *) : “Axiom 1 : Every number is succeeded by a number a’ différent from a Axiom 2 : There is exactly one number, called the first, which does not succeed any other. Axiom 3 : If the successors a' and b' of numbers a and b are equal, so are a and b. Axiom 4 :If a set of numbers contains the first number and with every number in it also contains the successor, then it contains all numbers. “
As F. Gonseth remarks: " With these conventions, all arithmetical operations that can be performed in the sequence of integers find their analogues in the sequence of even integers, and vice versa. Considered from a certain angle, the sequence of all integers thus appears equivalent to that of the even ones, although from another angle we know very well in what way the two sequences differ. This axiomatisation is accompanied by an analysis of 12
Ferdinand Gonseth Les Mathématiques et la réalité : Essai sur la méthode axiomatique, 1936. (Réédition: Librairie Scientifique et Technique Albert Blanchard, Paris,1974.) Extraits : Chapitre IV : le double visage de l'abstrait http://michel.delord.free.fr/gonsethg.pdf Chapitre VI : la nature du nombre entier http://michel.delord.free.fr/gonsethn.pdf
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the notion of number which is not without value: but its significance should not be overestimated. In any case, the axioms are in no way freely and arbitrarily formulated decrees, with the intention and ability to confer existence upon these entities called numbres. In particular (let us on this occasion remember our analysis of what is geometric), the notion of number has a whole aspect that is untouched by the axioms: it is precisely that aspect which in the exercise of thought is the most important; the one which relates to the idea of magnitude and, by analogy wih the specifically geometric aspect of spatial notions, could be named the specifically arithmetical or numerical aspect. The logical aspect is indeed not without value, because it is the one - which eventually becomes part of the basis for permitting something that seems obvious to those who know it, but is an important breakthrough of the human mind: the same abstract operation 15×17 yields the area of a rectangle of 15m by 17m as 15m×17m = 255m², and allows us to identify 17 coffee-breaks of 15 minutes as taking 255 minutes - which permits, in solving arithmetical problems, once the system of units and the operations to be performed are chosen, to calculate blindly, ignoring the nature of the units used, and yet be sure to arrive at the correct result. -which has permitted the creation of all the tools of formal computation. However, as Gonseth remarks, the idea of magnitude is lost here because, in this example, 1 is identified with 2, which plays the role of first number in 2N, whereas we know very well that they are quite different. In a sense, every number loses its meaning as magnitude simply because the unit is no longer perceived as a magnitude, i.e., a concrete number. It is at this point that we find the double nature of 1, which can be expressed (certainly not in elementary and not even in secondary school) in the form 1 = 1× 1 or 1=1× u or u=1× u, u being the unit; that is to say, the double nature of 1 conceived of as the unit and as the first number, or simultaneously as an abstract number and as the abstract representation of the unit magnitude. This difficulty, incidentally, was overcome historically as late as 1585 by Simon Stevin, who asserts, in the Euclidean form of a postulate "That the unit is a number"xv, at the very beginning of his "Arithmetic". However, this difficulty was overcome in 1585, only because it existed till then, as it sitll exists today, in the mind of the learner. However this may be, a mastery of computation with magnitudes presupposes not only its being taught as such, that is, as calculations using the International System of Measures rather than manipulatives, but also systematically in practical exercises about weights, lengths, areas, and volumes. Charles-Ange Laisant (co-founder of the journal L'enseignement mathématique in 1899, SMF’s president) replied to a parent who asked him which high-school to choose for his children: "Before even entrusting them to any establishment, you have the right and the duty to enquire about the spirit of its teaching, the methods it follows, and its working conditions, wthout having to be a mathematician yourself. Above all, do not let yourself be intimidated by the director , head-master , principal, or whatever the title might be, who might suggest that you are meddling with matters beyond your depth. Just two observations will give you an idea of how to pursue your enquiry. For teaching the metric system, some high-schools do not have a single measuring instrument: neither meter nor liter nor any weight. For teaching, geometry, […] Thus, if you are preparing to arrange for your child's attendance at a middle or high school, for example, ask to see the teaching materials for weights and measures, surveying instrumets, etc. If you are told that there is none of that in the house, take your leave at once and never go back13
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C-A Laisant, Initiation mathématique, Paris, 1910, dixième édition : http://michel.delord.free.fr/lais-init1.pdf
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To conclude, let us revisit the first lesson in Arithmetic and the System of Measures, by Brouet and Haudricourt Frères (Paris 1907).
Arithmetic Preliminary Notions 1. - The term quantity will signify anything that can be augmented or diminished, like a sum of money , a number of trees, or the height of a wall. 2. - The unit is a known quantity used to measure or "size up" all quantities of a given type. Ex.: In counting the tables in the class-room room or the trees in t he yard, the units are table and tree, respectively. 3. - A number is the result obtained by comparing a quantity to its unit. It is concrete if it shows the type of its unit, like 12 litres; it is abstract if it does not show the type of its unit, like 12. 4. - There are three sorts of numbers 1° The whole number, which contains only whole units : four francs: 4 . 2° The fraction, which contains only parts of the unit: twenty-five centimetres : 0m,25; - two thirds : 2/3. 3° The mixed number , which is a whole number accompanied by a fraction : three francs and forty centimes : 3fr. 40 - two and a third : 2 1/3 5. - Arithmetic is the science of numbers and calculation. 6. - Calculation is the art of combining numbers.
Two objectives regarding calculations with magnitudes Introduction to dimensional analysis: No orders of magnitude without magnitudes Michèle Artigue, a recognized specialist in mathematics teaching told us in 1982: We purposefully gave “idiotic” problems to students. The team of the IREM of Grenoble went even further in breaking the didactical contract by asking elementary school students nonsensical questions such as: “In a class, these are 4 rows of 8 seats, how old is the teacher?”; and we were shocked to observe that most of the elementary school studennts made an effort to solve these problems as though nothing was wrong ; and they did not chose the mathematical operations at random: the teacher was determined to be 32 years old14 . In my opinion there is nothing shocking about the procedure followed by the student15 who only acted as he was taught in accordance with the official curriculum for the past 30 years. This situation arises when: •
the student was taught only pure numbers, and therefore was not given the definition of the operations (which is possible only -- as was done in the chapter on the meaning of operations -- in the context of magnitudes), hence had no criteria by which to select operations or subsequently verify that the result made sense in terms of dimensions
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Michèle Artigue, Mathématiques : les leçons d'une crise, Sciences et Vie Hors Série N° 180 de Septembre 1992, pages 46 – 59. For a more acute critique, read : Michel Delord, Michéle Artigue et l'âge du capitaine, sept. 2003, http://michel.delord.free.fr/captain1-0.pdf 15
There are, however, several reasons to be shocked by this text, but they have to do with Michèle Artigue and the educational community, who have not yet reacted to it 13 years after its publication. In fact, while presenting itself as a critical summary of the New Math period, it puts the blame for the failure of that reform as usual "on teachers who where not sufficiently prepared". But it never mentions any responsability based on theoretical errors of its inventors, notably the abolition of "calculation on magnitudes", although they themselves had pointed to it as the central element of the reform. Moreover, when M. Artigue analyses student errors, this teaching gap does not figure among her possible causes. Instead she invents a new concept – le contrat didactique – whose main effect is to hide that absence.
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in addition, the heavy insistence on calculating orders of magnitude leaves him with this calculation as the only guide to his choice of operations,
the student proceeds in the following manner: he calculates 8+4=12, 8-4=4, 8×4=32, 8/4=2, 4/8=0.5, and since, in terms of order of magnitude, the teacher cannot possibly be 12, 4, 2 or 0.5 years old, her age must be 32 years. Not having a definition of multiplication, the student cannot know that any multiplication can be expressed in terms of multiplicand, a concrete number with a specific dimensional unit, and multiplier indicating the number of repetitions of the multiplicand, product having the same unit as the multiplicand. If he had known this, he would have been able to choose either the number of rows or the number of seats as the multiplicand, but he would have noticed that in either case the result of the multiplication could not have been the duration in years. Even if he had known only the rule (a procedural rule which is extremely important at the beginning of teaching) always write the multiplicand with its unit first: if you want meters in a multiplication, start with meters, he would not even have started writing the multiplication, since in writing down the multiplicand (either seats or rows), he would have known that he could not get years as the product. It would be a little tedious to develop the rules of dimensional calculation for each operation (and the appropriate and effective ways of expressing these for each level of teaching). Let us state explicitly that the students learn more about the definition of an operation when they can form a mental image of it16 . We will chose the definition of multiplication found in Brouet et Haudricourt Frères, Arithmétique et système métrique Cours Moyen, LibrairiesImprimeries réunies, Paris, 1912 noting that all of the “other multiplications” (for example 3€/m × 5m= 15€, 3m × 5m = 15m2 , m2 × 5m = 15m3 …) should first be taught in this format.
Multiplication 68. Multiplication is an operation whereby one repeats a number called multiplicand, a number of times indicated by another number called multiplier. The result is called product. […] 17 70. The multiplicand and the multiplier are called factors of the product. 71. Mult iplication is indicated by the sign x (multiplied by) which is written between the numbers to be multiplied: 8 × 5 (8 multiplied by 5). 72. Multiplication is only an abbreviation of addition. 73. The multiplicand is always a concrete number, that is one which describes a specific object, such as trees, meters, dollars, … 74. The multiplier is an abstract number that indicates only the number of times that one repeats the multiplicand. 75. The product is always in units similar to those in the multiplicand.
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Of course, like everyone else, a student has mental images of mathematical notions. However, the teacher's role is not to control these images, which are in the domain of the student's present and future inventiveness, but to verify that they are compatible with subject knowledge. The student has every right to think that 2 is red and 3 is yellow : the teacher's main task is to make sure the student knows 2+3=5 and 2×3=6, etc. 17 Item 69 gives definition of multiplication based on proportionality: Multiplication can also be defined as follows: 69. – Multiplication is an operation whose aim is to find a number called the product, which is to the multiplicand as the multiplier is to unity.
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Higher mathematics and elementary arithmetic The statement of BO in 1970 [previously cited] “Sentences such as 8 apples + 7 apples = 15 apples are not part of mathematical language” is a stupidity which primarily has the function to prohibit writing that 2m + 3m = 5m, or 2m = 20dm and in fact any calculation of units. It is more important to understand the reasons for this prohibition. In the “axiomatic” system which these authors posed: i) it is not permissible to introduce a concept without its pure mathematical definition, ii) there is a sequence in teaching knowledge - that’s true - but this order must be the axiomatic order; In this system the simplest algebraic structure which can contain expressions such as 2m or 2m + 3m = 5m, is that of a one-dimensional vector space, which follows logically but much later than the concept of whole numbers. And since it is impossible to introduce the notion of a vector in Grade 1 when one is learnig whole numbers, they prohibit the learning of concepts such as 2m + 3m = 5m or 2$ + 3$ = 5$, perfectly understandable at the intuitive level by everyone … except teachers. On the contrary, it is extremely important to: i) to learn, if one follows the question of unit, starting at the earliest age, to write down identities of the type 2m + 3m = 5m, 2m2 + 3m2 = 5m2 , 2m3 + 3m3 = 5 m3 , 2×3m= 6m, 2m×3m= 6m2 , 2m×3m2= 6m3 certainly without making any allusion to their character as algebraic identities but only justifying them to calculate measures of length, surface area and volume. In doing this, one introduces structures of thought which one will find 5 or 10 years later because they have the same syntax of formal monomials 2x + 3x = 5x, 2×3x= 6x, 2x×3x= 6x2 , 2x×3x2 = 6x 3… ii) More generally, never hesitate to introduce a concept if it can be understood at an intuitive level, even if it is not possible to provide a formal mathematical definition at that moment, because this training will serve as an intuitive base for the subsequent introduction of its more formal mathematical definition. One example is the following property of measurement. If the unit u is divided by a number, the measurement is multiplied by that number. This property, like inverse proportionality, and properties which use two variables with a constant product, is an example of inverse variation which cannot be formalized mathematically until much later, probably at university. The reason for abandoning the notion of inverse proportionality and its applications come mainly from the fact that there are no inverse linear functions for which addition would have been characterized by f(x)+f(y) = (f(x)+f(y ))/f(x)f(y) . Conversely, another example consists of not refusing to add additional concepts to those which have already been understood at the intuitive level. The most interesting example is to use the semipolynomial concept of positional notation to introduce at grade 8 or 9 formal polynomials as an extension of the concept of whole numbers. So this comes down posing a polynomial operatins exactly as whole number operations18 , because since 3021× 201=607221, one can deduce that (3x 3 +2x 1 +1)×(2x 2+1)= 6x 5 +7x3 +2x 2 +2x+1.
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This introduction to polynomials had been used in France up to the sixties (last example : M.Monge et M. Guinchan, Mathématiques, classe 4e, Librairie classique Eugène Belin, 1965) but you can find it in USA in Dolciani, William Wooton, Edwin Beckenbach Algebra1, Houghton Mifflin Company, Boston, 1974, page 300.
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Numerical calculation: from arithmetic to field axioms Admittedly the mathematics teaching of today has left behind the worst excesses of the past: a New Math which integrally combined the teaching of mathematics with that of its foundations19 . However, a fundamental legacy of the New Math remains, which is, in fact, the direct aiming at the most abstract notions20 , as still found in the teaching of elementary arithmetic. Sure enough, all numerical calculations in primary school are mathematically related to the axioms of fields or commutative rings, but the latter: - are axioms, i.e., they define the minimum of rules necessary for all calculations, - only involve two operations (+ and ×) whose only bond is the distributivity of × over +, - are expressed as algebraic identities (i.e., written in letters). Even if understanding the importance of these axioms is a conceivable objective of secondary teaching, they cannot be used to guide elementary teaching, for the following reasons. a) Symmetry of equality? The written form of the axioms presupposes the symmetry of the relation "=" as well as the fact that this sign separates two different expressions of the same number, which is one of the aims of primary teaching, but not its point of departure. The spontaneous meaning given to this sign by the student (if he had been taught the algorithms of the arithmetical operations) is as follows: it separates the given numbers (on the left) from the result of the operation (on the right), and symbolises the mechanism which permits the passage from left to right. This is to say: a priori expressions like 15=7+8 or 4+2=2× 3 mean nothing to the young child, or worse, they do not mean what the teacher thinks in forcing the student to write them. b) Two arithmetic operations or five? Children have to learn four operations (+, -, ×, :) and not just two. Furthermore, the following profound observation by Ron Aharoni has to be taken into account: "... in fact there are five operations. Beside the four classical operations there is a fifth one, more basic and important: that of forming a unit." In this sense, 1is essentially not the neutral element of mutliplcation, "multiplying by 1 does not change the result", but means first and foremost that every whole number is a multiple of the chosen unit, which is denoted by 1.
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Some have even gone so far as to demand, set theory not being fundamental enough, that the theories of categories and filters be taught. See, for example about filters, Bulletin de l'A.P.M.E.P. N° 302 (February 1976) and Henri Cartan’s statement about teaching filters in high school : http://michel.delord.free.fr/cartan75.pdf 20
One could speak of "conceptual teaching" in the sense of trying to teach "concepts" directly, a problematic approach which has now spread to all subjects: "The disciplines of primary school are labelled - as a sign of the uniformisation of what is now called 'the educational system' - from Kindergarten onward in the same manner as in secondary school. Just as the primary curriculum no longer includes drawing or gymnastics, but fine arts and physical education, it no longer contains natural science, not to mention natural history, but biology instead. The uniformisation of school has shattered the pedagogical paradigm of a progression from the simple to the complex. In view of their anticipated future schooling, students are immediately initiated, at their most tender age, to the complexity of the subjects they are to master, in order to be able to market them later, in their adolescence, as well as possible. Exit the object lesson conceived as a lesson in observation. Beginning in primary school, the student no longer gets acquainted with 'objects' but with concepts (emphasis in the original - MD) : no with longer the digestive system but with digestion ; no longer with the principal functions of living, but with the construction of the concept 'life'. As to the descriptive classifications of the Three Kingdoms of Nature, which used to constitute the body of the science course from elementary to middle school, they simultaneously lose their pedagogical and epistemological legitimacy." Pierre Kahn, De l'enseignement des sciences à l'école primaire; l'influence du positivisme, Hatier,1999. http://pst.chez.tiscali.fr/svtiufm/positivi.htm
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c) Distributivity = associativity? The connection between addition and multiplication, as taught, does not correspond to the standard distributive law: - distributivity itsef appears essentially in the form "to multiply a sum by a number, each term of the sum must be multiplied by that number" , which means (i) only the direction from right to left in a(b+c) = ab+ ac, (ii) without limitation on the number of terms. - in primary teaching, it is more related to the following property of multiplication, "to multiply a product by a number it suffices to multiply each factor of the product by this number" , which in itself is an asymmetric expression of its associativity [id est k × (a × b) = (k × a) × b]. In the language of axioms this sounds more like the non-distributivity of multilplication with respect to itself. d) Multiplication and division. Since division does not occur among the axioms for fields or rings, any reference, for instance, to the relations between division and multiplication cannot occur either. However, the following three properties (the last one being the simultaneaous application of the first two), which are not part of these axioms but apply to any division problem: (i) if the dividend is multiplied by a certain number, so is the quotient, (ii) if the divisor is is multiplied by a certain number, the quotient is divided by it, (iii) if both dividend and divisor are multiplied by the same number, the quotient does not change21 , are nevertheless fundamental, because they are the basis for understanding 1) the simplification of fractions, and 2) the justification of the division algorithm for decimals. e) Commutativity of multiplication. Even if one of the axioms is directly necessary in primary teaching, its status is not the same as the one it has in a purely mathematical perspective. Taking for example the initial purpose of teaching the commutativity of multiplication, (i) it will not be taught in the form a × b=b × a (let alone with quantifiers) but the form "a product of two numbers does not change if you change the order of its factors" (ii) its principal uses are - to be able, in a multiplication problem, to put "on top" the number with the most digits, - to check a multiplication by inverting the order of its factors. f) Arithmetic and language. None of the above-mentioned rules of calculation are given in an algebraic formulation but, on the contrary, in everyday language. The principal reason for this is that students are apt to miss their full wealth of meaning -- this is a real risk -- if they have only a formal understanding of them, but above all, that any real grasp of abstraction must always pass through an exp lanation in words. Consequently, any mastery of mathematics, at whatever level one might consider, depends on a mastery of language, which in turn is possible only through a particularly solid command of the grammar which enables the expression of its subt leties. This dependence is all the more real as mathematics is, among all subjects, perhaps the one requiring the greatest degree of precision in the formulation of its reasoning and its properties. 21
And the remainder is multiplied by this number. This is the basis of an exercise which shows whether a student has really understood division of a decimal by a decimal : Record the remainder of the quotient 2.3753 / 0.7 based on 1/10 from the written long division [without further calculation].. The answer is, of course: 0.653 since 0.7×3.3 + 0.653 = 2.3753.
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"Our language expresses, by its inflections and even its word order, indefinitely subtle nuances. The minutest of these nuances can viciate any mathematical reasoning where a straight line must be followed and the slightest deviation is barred. To understand these nuances, one must have learnt to sense them ; one must have acquired a long acquaintance with them, to grasp them at once, without hesitation or effort. A child understands sentences as monoliths, so to speak, and would write each of them as a single word if it could. Each word is like a centre of association of ideas, a torch which lights up a whole region of consciousness; the different words of the same sentence shine simultaneously ; their radiance intermingles , the fields they illuminate overlap, so that one cannot say from which of them a particular point receives the most light. This is how to understand myopian vision, which makes the various points of an object appear as blotches spilling over onto one another, resembling those admired in certain modern paintings. It is this kind of blurred illumination which is normally taken to be the meaning of a sentence. Many people, even adults, do not ask for more ; the most refined among us are content with it nine times out of ten ; this manner of understanding language is, in fact, sufficient for most of the tasks of everyday life. Every sentence, by the simple interplay of associated ideas, suggests the appropriate movements ; when we are told to turn right, the muscles which make us turn right contract all by themselves. That is enough to get us through life. But it is already too little for most civilised people; it is altogether insufficient for something as subtle as mathematical reasoning. Through this delicate rolling-mill, monolithic sentences will not pass ; it needs to be presented with material less coarse, reduced, as it were, to small pieces by verbal analysis.. For those inexperienced in such gymnastics with words, [the expressions] 'which multiplies B' and 'which B multiplies' do not primarily represent a relative pronoun as subject and object, but some vague, incomprehensible notion about multiplication ; but this vague notion is of no use to the mathematician." Henri Poincaré, in Les sciences et les humanitésxvi , Paris, A. Fayard, 1911.
Cabanac, le 31/05/2006 Michel Delord
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Brief history of French national arithmetic demands for grade 1 and 5 A slow but sure comeback to Middle Ages' pedagogy 1882-1970 1st grade22 : Numeration up to 100 and 4 operations (multiplication and division by 2, 3 and 5 only) 5th grade: Mastery of all the four operations on whole numbers and decimals. Reform of 29 January, 1970 (Modern Mathematics) 1st grade: Numeration, addition 5th grade: Multiplication algorithm for decimals, but no more than "exact division of a decimal number by a decimal number", and "limited to simple numbers". Approximate quotients are no longer found in the exercises, but "The meaning of "exact to an error of
