For the Teaching of Mathematics Volume Three

For the Teaching of Mathematics Volume 3 Elementary Mathematics

Caleb Gattegno

Educational Solutions Worldwide Inc.

First published in 1963. Reprinted in 2011.

Copyright Š 1963-2011 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-337-1 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

In This Series 1

What is Modern Mathematics?

by Gustave Choquet

2 For the Teaching of Mathematics

by Caleb Gattegno Volume 1 Part I Mathematics and the Child Part II Pedagogical Discussions Volume 2 Part III Psychological Studies Part IV On Films Volume 3 Part V Elementary Mathematics Volume 4 Part VI Miscellaneous Topics Part VII Mathematics Teaching and Society Book Reviews 3 Talks for Primary School Teachers

by Madeleine Goutard 4 Mathematics and Children: A reappraisal of our Attitude

by Madeleine Goutard

Table of Contents Preface ........................................................................ 1 Part V Elementary Mathematics .................................. 7 1 Notes on Monsieur Cuisenaire’s Invention...........9 2 Number and Color ............................................. 13 3 Arithmetic with Colored Rods ........................... 21 4 Arithmetic and the Child ................................... 25 5 Georges Cuisenaire’s Numbers In Color ............ 31 Conclusion……………………………………………………………40 6 The Study of Arithmetic with the Help of Color Associated with Length ...............43 7 Learning Mathematics—A Practical Solution ..... 57 Substitutes for Actions………………………………………..…58 Color and Number…………………………………………………59 Fractions and Classes of Equivalence……………………………………………………….….61 Naturalness of Algebra…………………………………….…….62 8 Theoretical Remarks on the Cuisenaire Material.......................................... 65 The Nature of Mathematical Activity……………………….65 The Formation of Arithmetical Activity……..…………….70

9 Introducing the Concept of the Set .....................79 Introduction………………………………………………………….79 Characterization of the Mathematical Activity…………80 Structures and Relationships………………………………….81 Arithmetic……………………………………………….……………83 Conclusions…………………………………………….…….………87 10 Notes on a Radical Transformation in the Teaching of Mathematics ...................... 89 11 Thinking Afresh About Arithmetic .................. 101 12 Reality and the Learning of Mathematics........ 107 13 A Matter of Relationships ................................117 Color Relationships and Values………………..…………...121 14 Mathematics for All ........................................ 125 15 Note for Administrators ................................. 131 16 Multivalent Materials ..................................... 135 17 The Cuisenaire Material is Not a Structural Apparatus............................ 153 18 Why My Books Are as They Are ...................... 159 19 The Place of Color in Mathematics Learning... 165 20 Discovering Cuisenaire .................................. 173 Appendix.................................................................. 179 The Sources of this Volume ......................................189 Index........................................................................ 191

Preface

This volume mainly contains articles outlining the development of my ideas concerning Cuisenaire’s discovery of a new basis to the learning of arithmetic. After more than ten years of work I am still of the opinion that Cuisenaire’s contribution to elementary mathematics education has no equal in the whole of the history of this subject, except perhaps for the introduction of the decimal notation including the invention of the zero.

This preface is not the place for as full a justification of this opinion as may be needed by skeptical readers, but I can at least outline the depth and the breadth of the revolution to which his discovery has given rise.

A primary school teacher, after his retirement from public service in Belgium, published eleven years ago a booklet in French called Nombres en Couleurs. A number of visitors saw his pupils at work in the classes of Thuin, but his idea was nevertheless unknown to people deeply involved in mathematics teaching investigations as little as ten or so miles away. As Cuisenaire had retired from his post of Director of Schools for

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For the Teaching of Mathematics – Volume 3

Thuin, he was not in a position to promote his ideas except through writing. His book, couched in the language of the now dated “activity school�, could only touch teachers who were believers in the principles propounded by that school. The most valuable aspects of his contribution could easily be lost for most teachers, belonging as they do to a large variety of educational sects. I consider it fortunate for both Cuisenaire and myself that, rather than simply talk to me or merely give me a copy of his book, he showed me a group of children working in accordance with some of his ideas.

In 1940, I had tried my hand at teaching mathematics to infants, but my efforts met with complete failure. Three years later, I gave up hope of ever achieving this elusive end. But I was still concerned with this experience, and what Cuisenaire showed me put me right at once. I knew immediately that my preparation as a mathematician, a psychologist and an educator could be of use in propagating what I had just met.

As a mathematician, I at once started on the presentation of classical elementary mathematics topics in modern terms, introducing, in particular, classes of equivalence and equivalent expressions in the fields where they belonged. This made it possible to transform beyond recognition the situation with respect to subtraction, long division, fractions and mental arithmetic.

Although a large number of reformers of mathematics education today propose as fundamental changes the introduction of sets and logics in elementary mathematics, my view is that the

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Preface

creative side of mathematics lies somewhere else; therefore, what we should do is to widen the horizon and increase insights, hence the stress in my books on transformations and on maintaining the initiative of the learner. For example, the study of algebra should be undertaken before arithmetic (this is made possible by the manipulation of the colored rods); all operations should be met at the same time (as early as when studying integers up to 5); whole numbers should be recognized as the class of their partitions (using addition as a constituent operation), instead of the classical definition as the class of idempotent sets; fractions should be considered as operators or ordered pairs belonging to classes of equivalence which are the rational numbers involved in the operations; powers and logarithms on various bases should be seen as an interpretation of an isomorphism between addition and multiplication; an algebra (that of polynomials) should be recognized as the backbone of calculations in different bases of numeration, and so on.

As a psychologist, I have been able to study the way in which mathematical mental structures are formed, and how the various notions can be linked with specific actions upon the rods so that it is possible to produce from scratch a large number of entries into mathematics avoiding the linearity of the development of the subject generally proposed. I am now confident that the major obstacles to mathematics learning have been removed for most learners and that the vast majority of our present and future school populations will be good at mathematics, provided their teachers learn to work with these materials.

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As an educator, it was my task to develop at least one complete syllabus adapted to the new method so that children could be taken step by step from their first day at school with no formal mathematical knowledge to the most realistic level of maximum achievement. This I did in various ways in my books for pupils, in another way in my filmstrip, in yet other ways in my publications for teachers and in courses that have been recorded on tape in Buenos Aires, New York State (New Rochelle, Brentwood, Rochester), California (Menlo Park and Santa Rosa), in Canada (Montreal, Quebec City), England (Jordans, Slough and Reading). The table at the end of this book, which is published in English for the first time here (a Spanish version appeared in Madrid in March 1961, and in Buenos Aires in November of the same year), shows that there are indeed a number of routes from which a selection can be made if one wants to produce syllabuses.

What seems much more important is that we now have a widely tested course of study which pupils of ten or eleven years of age can easily understand and master, whereas traditional methods require four or five more years and seldom lead to understanding—even more seldom to mastery.

There is no doubt for me that the work that started when Cuisenaire came on the scene is the most hopeful sign for a true adjustment of educational techniques to the state of our knowledge of mathematics today and to the subordination of teaching to learning.

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Preface

I am deeply indebted to Jeremy Steele, who assisted me in the preparation of the scripts for publication. Improvements on the originals were permitted only where the substance was not touched.

Reading, May, 1963.

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Part V Elementary Mathematics

1 Notes on Monsieur Cuisenaire’s Invention

This extract first appeared in November 1953, in Le Moniteur des Instituteurs, Tamines, Belgium.

The idea is so marvelously simple that it escaped the attention of everybody, having to wait for exceptional circumstances before it was eventually seized upon. The genius of Archimedes was astonished by the pressure experienced by his body in the bath; Newton realized that objects fell, which everybody could see all the time but which they never really stopped to think about.

Georges Cuisenaire saw that the basis of mathematics was relationship, or relationships, and he produced a material owing its existence to this fact; he colored pieces of wood, or rods, using similar or differing tints according as their lengths were, or were not, clearly related. If one number was twice another, their colors were very similar (shades of red, green, and yellow). If, on the other hand, what they had in common was not so simple, the colors were strikingly different (red and black, or yellow and blue, and so on).

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Rods of different colors, and whose length increased in units of one centimeter each time, from one to ten, in sufficient quantities, permit us to elaborate the development of most of the questions in primary and secondary school syllabuses for arithmetic and algebra, and for certain questions in metric geometry besides.

Let me take an example of where the use of the rods can bring substantial progress in a very short time, an example of an actual lesson, chosen from among many.

Pupils who have become familiar with the rods by playing with them examine relationship by exactly matching the length of one rod with two other smaller rods, both of the same color. In this way the fact is established that two whites make a red, two reds make a pink, two greens make a dark green, two pinks a tan, and two yellows an orange. From this situation the expressions “double” and “half” are first extracted, then is represented by placing the smaller rod alongside the longer.

Thus there are five ways of representing a half: one on two, two on four, three on six, four on eight and five on ten.

Since the rods all have the same cross-section they can be arranged one next to the other either horizontally or vertically.

The following relationships can thus be perceived: if the various ways of representing are arranged in increasing order in terms of the base rods it can be seen that the increase in both vertical and horizontal rods is constant, the increase in magnitude of one set of the rods being twice that of the other set, that is, 2, 4, 6, 8, 10, and 1, 2, 3, 4, 5.

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1 Notes on Monsieur Cuisenaire’s Invention

If pairs of rods are compared at random, it can be seen that the smaller rods stand in the same relationship to each other as do the larger ones: 2 is to 5 as 4 is to 10, and so on.

Again, if the role of the rods is reversed, different ways of conceiving of the double, or twice, are obtained. Similarly, one can start again with a third or a quarter, and so on. This provides a much more real and flexible knowledge of fractions than does the usual static method.

Let us now take the white (1 centimeter) and the red (2 centimeters) rods. By placing one on the other, is formed when the red is the base, and 2 when the white is the base. If the green rod is added to the series, so that a white, a red and a green rod can be seen, when the white is taken as a base, the red becomes 2 and the green 3; if the green is the base, the white is and the red (which is always twice the white) is ; and if the red is the base, the white is and the green is .

When the pink rod is added to the series and the white again taken as the base, then the red is 2, the green is 3 and the pink 4.

If the pink is the base, then the white is J, the red (which is (also because two reds equal one always twice the white) is pink) and the green, which is always worth three whites, is (also because ).

By increasing the series each time (through the addition of the yellow, the dark green, and the black, and so on, in order), the pupil acquires a wealth of mathematically correct experience in fractions offered by no other method. It is both abstract and

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concrete; it shows what is invariant in situations and shows clearly and simply what the variables are.

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2 Number and Color

First published in New Era for Home and School, London, January 1954.

The advantage of using color in teaching number becomes very clear to those who have seen Georges Cuisenaire’s material used either by children or adults. Most mathematics teachers use colored chalk, and its value, particularly in the teaching of geometry, is generally recognized. Cuisenaire, however, has discovered that color relationships are a very adequate answer to the problem of bridging the gap between concrete experience and the abstractness of mathematical notions, and has designed a material where color becomes an integral part of the learning process in arithmetic.

Before proceeding further, let us introduce the material and its inventor. Monsieur Georges Cuisenaire is the Director of Education in the town of Thuin in Belgium. He is well known as an enthusiastic and progressive educator and his widely read books represent a very sound contribution to educational ideas.

He has for years been concerned with the problem of why, in so

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Part V Elementary Mathematics

many schools and with so many children, the arithmetic taught fails to produce the delight, the efficiency and the accuracy of thought which are, for the mathematician, inherent in the science of number. He finds the answer in the fact that we have failed to ensure that information acquired in everyday experience becomes a dynamic set of relationships, because we have failed to see how it could be done. We have therefore presented the child too soon with abstract word and concepts and in an unnatural manner. A diagnosis is interesting but it needs to be followed by a cure, and this Cuisenaire has achieved through experiments with a semi-abstract set of concepts which he has embedded in a material and a method.†The arbitrary connections between these rods are learnt in a matter of minutes by anyone who can distinguish between colors.* There is no need for all the connections to be learnt at once.

Shortage of space makes it impossible to give here any detailed account of the ways in which these rods can be used, which has, in any case, been adequately done in the text provided by Cuisenaire himself entitled Numbers in Color. What should, however, be said is that these rods are the first fully satisfying

†Here follows a description of the Cuisenaire rods. * For the color-blind, the various lengths and a vague impression of differing tones of color provide the basis for identifying the rods both as numbers and as colors.

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2 Number and Color

material to be devised for teaching arithmetic†, for the following reasons.

Technically, from the fact that the rods are not numbered or divided, each possesses an individuality that eliminates two major difficulties in the introduction to number—the supremacy of 1 and 10. In reality, any number is equal in status to any other. The importance we give to the procedure of forming numbers by adding 1 is justified only insofar as it is convenient on logical grounds, and the importance given to 10 arises simply from our way of writing numbers. These are in fact only two of many possible approaches to the understanding of number. We form new numbers by operations such as division (fractions), by finding square roots or by considering the fraction of a fraction in order to obtain a fraction, and so on. Progress in mathematical experience very soon takes us beyond the limited field in which we count in units and stress the tens, and it is mathematical experience that the rods provide. In forming a number by placing two or more rods end to end we obtain the additive properties of whole numbers. Thus 11 is 10 + 1, or 1 + 10, or 9 + 2, or 2 + 9, or 2 + 3 + 6, or 2 + 3 + 4+1 + 1, and so on (called decompositions of 11). This property belongs to all numbers.

Moreover, with the rods it becomes technically possible to introduce the operations of arithmetic in pairs at one and the

† Seguin, Montessori, Stern and others have, of course, devised material from which children can learn much, but it is less flexible and, to my mind, less mathematical than Cuisenaire’s.

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Part V Elementary Mathematics

same time, addition and subtraction on the one hand, and multiplication and division on the other. This is, of course, as it should be. Usually, in teaching arithmetic, there is a tendency to introduce each operation separately and to keep them separate for some considerable time in the belief that the difficulties will thereby be minimized, although work with infants and juniors proves the contrary to be the case.

Once a number has been formed with some of the rods the pupil can then make all the ‘decompositions’ he can think of, as described above. He will find that while a particular number will have many decompositions made up of various numbers of rods, there is a minimum number of rods that any decomposition of this number must have—thus, for 35, this minimum number is four (3 × 10 + 5); for 36, four again (3 × 10 + 6, or 4 × 9). These special decompositions can be singled out from the set of decompositions and envisaged as providing complementary numbers: for example, 6 + 7 = 13, 5 + 8 = 13, 10 + 3 = 13, and so on. By removing a rod from each row except the top in a decomposition table, the children can try to discover the value or color of the one removed, which is, in fact, subtraction from 13 of any of the numbers up to 10. Variations on this procedure will provide exercises that will familiarize the class with the two inverse operations of addition and subtraction.

It should be noted that the fact that a set of decompositions does not contain a row all of one color indicates that the number concerned is a prime number, and that when rows of one color do occur they provide the basis for the factorization of that number. For instance, possible formations of 12 include four threes, three fours, six twos and two sixes, and these are 16

2 Number and Color

presented at one and the same time in one situation. From this situation, we can learn the value of the products 3 × 4 = 4 × 3 = 6 × 2 = 2 × 6, and also that 3 is a quarter of 12, that 6 is half 12, and so on.

Hence we do not learn multiplication tables and we do not learn the products as a connected set, but we come to know them through manipulation and study of the decompositions of those numbers up to, say, 100 which contain rods of one color only.

Though we may never think of products as tables we can answer without hesitation that 8 × 6 is 48, that 48 is 8 × 6, 4 × 12, 3 × 16, and so on.

This progressive study of arithmetical facts within learning situations is one of the most distinctive and most valuable features of the Cuisenaire material. Children are not required to learn this after that, but are left to discover what they can in a highly mathematical situation formed of the interaction of relationships.

Suppose, for example, that we want to examine what is involved in subtracting 17 (or 27) from 23 (or 33). By putting side by side the two numbers formed with the minimum number of rods, that is 1 × 10 + 7 and 2 × 10 + 3 (or 2 × 10 + 7 and 3 × 10 + 3), we find that what is needed to bring the one (17 or 27) up to the other (23 or 33) is the same in both cases, and that it is what is needed to bring 7 up to 13, that is, 6. Observation and study of such situations leads to a clear understanding of what is meant by subtraction with borrowing, and there will be no new difficulty in passing from subtractions such as 27 – 13 to those

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requiring that we discard in the two numbers as many tens as are common to both. In writing down the operation the pupil will visualize the two given numbers as individual entities and will write down the answer as he sees it. As an extension of this, in the case of, say, 467 – 298, the child trained with the rods will first see that this operation is equivalent to 267 – 98, then that 2 is needed to raise 98 to 100 and that the answer is therefore 167 + 2 or 169. It is in this way that I, as a mathematician, would work, and it is the way taken by classes trained by Cuisenaire’s method. Equi-addition of 2 seems natural in such a case and is naturally used. To take a further example, if the child were confronted with, say, 467 – 259, his experience with the rods would lead him to consider first 267 – 59 and then to concentrate on 67 – 59, since 59 is known to be less than 67. The answer for him would be 8 + 200.

These examples will, I hope, suffice to show that through using the rods the child can acquire mathematical attitudes even in the first years of his infant and junior school studies. It should, however, be added that the rods can also be used with profit up to the sixth form at the secondary school level and by students in training colleges. I have myself used them, with pupils from 15 to 19 years of age, in lessons on arithmetical progressions, and permutations and combinations, which have removed the haziness that is too often still present in our university classes.

Cuisenaire was fully aware that his material must be adequate for the passage from the concrete to the abstract and that his pupils must not become dependent on the size or color of the rods. His technique consists in providing as soon as possible for the writing of the operations performed, and in replacing the 18

2 Number and Color

rods first by other similarly colored symbols printed on cards and then by the usual notation, thus ensuring the transfer of the mental activity with apparatus to work with purely abstract relationships.

To begin with, the figures and the signs for operations are learnt merely as conventional signs by which the children can indicate rapidly what they are doing. The various rods are put down as 1, 2 . . . 10 instead of by their colors; the fact that they are placed end to end is represented by the sign + between each rod and the next; comparison of rows in order to discover what needs to be added is represented by —. If in any given decomposition there are several rods of one color, they are counted and written down as, say, 3 × 5 + 7 + 2 + 1, standing for three yellow ones, followed by a black, a red and a white. When the child writes 12 = 4 × 3 he means that the length 10 (orange) + 2 (red) can also be obtained by means of four green rods. A pink and a green rod are placed crosswise one over the other, by convention another way of representing this situation. The new convention for multiplication soon becomes familiar and can serve for the special study of products. This convention, in its turn, is replaced by another, the products being represented on cards by a colored symmetrical figure formed of three circles, two of them cutting the third. The middle circle is uncolored whereas each of the others bears one of the colors of the rods. This provides us with a table containing first the products of family colors and then those of different families. All the products are thus formed and the factors are seen in the colors. Counters bearing the product in figures can be placed on the middle circle. In some cases four different factors are represented: for example 12 = 6 × 2 = 4 × 3.

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Cuisenaire’s method has been elaborated so as to lead to truly mathematical thinking and efficiency and the pupils with whom it is used prove clearly every day that the claim is justified.

The rods and the colored signs may have potentialities still to be discovered. It is already known that the children who use them thoroughly enjoy their work and achieve a speed and accuracy surpassing all expectations. What they learn they know for good because it is their own discovery, not merely something they have been told. The extent of this knowledge far exceeds what is usually achieved except by exceptional children.

I have no hesitation in saying that Cuisenaire’s material has solved the problem of teaching arithmetic and should be made widely known in order that there may be an end to the painful struggling of children learning and to the frustration of those teaching.

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3 Arithmetic with Colored Rods

This article first appeared in The Times Educational Supplement, Friday, 19th November, 1954.

Georges Cuisenaire is the director of education for Thuin, in the Belgian province of Hainaut. He is well known as a progressive primary school teacher whose books have been a valuable aid to his colleagues in overcoming some of their most resistant difficulties. Some 25 years ago he turned his inventive genius to the study of the difficulties encountered in the learning of arithmetic, and his findings are now available to all teachers in the form of a fully elaborated method.

All those who have visited his schools at Thuin know that this method is not just a set of ingenious hints, well arranged, but that we are confronted, for the first time, with an outstanding solution to the difficulties met in the teaching of arithmetic, a solution for which the skeptical teacher is not prepared but which nevertheless is truly operative. Although words can be but a poor substitute for observation of children at work with the method, we shall attempt here to consider some of its aspects.

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The essential idea is that, by using color functionally, the gap between the concreteness of everyday life and the abstractness of arithmetic can be easily bridged, and that children can think arithmetically when they are considering obvious relationships exemplified in the colors. Concrete action is by its nature slow, whereas the swiftness of vision is very close to that of thought. If we systematically substitute color relations for the abstract relations of arithmetic, the child who sees and talks about the former is not distinguishable from the child who reasons about them, since what he is doing is expressing verbally what is taking place in his mind. Moreover, these clearly seen relations generate a mental activity which gradually takes the child ever further along the path of awareness of what can be done with numbers. This is what is so amazing in the classes using the Cuisenaire method.

The method comprises three consecutive approaches, consecutive, that is, at each stage of the learning process, not in relation to age. The first involves the use of colored rods, the second is related to colored cards, and the third to ordinary written work. The rods are so colored that some of the relationships needed are very easily seen; thus there are three rods whose basic pigment is red, three whose basic pigment is blue, two with yellow, one unstained, and one black. As the unstained rod has a length such that all the others are multiples of it, when it is chosen as unit the others measure 2, 4, 8 (red), 3, 6, 9 (blue), 5, 10 (yellow), and 7 (black) times that unit, here a centimeter. The rods have equal sections: 1 square centimeter.

One gesture that almost all children will make is that of putting two rods end to end, and this is basic in the work with the rods 22

3 Arithmetic with Colored Rods

for it allows of the passage from any length to any other. It is obvious that if we make with the rods all kinds of arrangements whose lengths are equal to a given length, we need only to read the colors to state what we see. If we then transcribe what we see, using Arabic numerals instead of the colors, we find that we can produce statements such as: 13, 6 + 7, 1 + 2 + 4 + 6, 8 + 5, 3 + 10, 2 + 9 + 2, 1 + 10 + 1 + 1, and so on, where the plus sign merely indicates end to end placing.

Now—and this is Cuisenaire’s stroke of genius—children can meet all the arithmetic operations in each of these sets of decompositions, and if they are made aware of them they will move from one number to the next, seeing them not only as being obtained by the addition of one unit, but as an aggregate of properties that are characteristic of each number, whereas the procedures are generally independent of the numbers. Thus, while 9 provides us with only one single-colored line (formed of rod 3), 12 provides four such lines, and 11 and 13 provide none.

We can, however, always attempt to form a given length with rods of only one color and we then see that there are only two possible situations. Either the given length can be obtained, or it must be completed by using a rod of a different color, a rod which is always smaller than the others. We thus discover how addition (putting rods end to end) becomes multiplication, and when the process is not completed, yields division with remainder.

Another important feature of the method is that operations are paired and the child learns at the same time facts of addition and subtraction, of multiplication, and fractions. At the age of 7

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Part V Elementary Mathematics

Cuisenaire’s pupils have no difficulty in working with fractions such as , , , , , , etc., and they understand fully what they are doing. Their confidence, and the swiftness and accuracy of their work are so impressive that the many visitors to Thuin find it difficult to believe that they are as young as they actually are. Long vacations do not seem to affect the skills acquired, and we can no longer assess them in terms of memory: it is true understanding that the children have found for themselves.

Cuisenaire’s own pupils, and those who are using his method, are equally skilled with the rods, the cards (which bear the same colors as the rods but are no longer capable of being manipulated), and the ordinary written signs. They can answer in a fraction of a minute at the age of eight questions such as the following: . This rapidity is not achieved as a result of undue time or effort. On the contrary, the pupils quickly come to need only 15 minutes of arithmetic a day to remain proficient, and the time thus made available can be used for other subjects.

Since we must teach arithmetic to all children, it is a satisfying thought that it can now be done pleasurably and efficiently, and that we can eliminate frustration in a subject which has for generations challenged the minds of teachers.

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4 Arithmetic and the Child

This article first appeared in Florence in Il Centro, June-July 1955, 3rd Year, No. 5.

To mark the second anniversary of my visit to Thuin, I should like to present some variations on the theme of Numbers in Color. To visit Thuin and to watch the inventor of so simple and so fertile an idea as Numbers in Color, Monsieur Georges Cuisenaire, at work, is such an overwhelming event in the life of an educator—overwhelming, at least, in my life—that the whole course of my existence was changed. I could not now live without integrating with my own vision of education all that this Belgian schoolmaster had revealed to me—which went far beyond a pure and simple teaching of arithmetic. Nor could I live without devoting myself to understanding better, to extending further, to developing, and to propagating the benefits of this present which had been freely given to me.

Monsieur Cuisenaire was so generous with his time and his talents that he showed me for an hour or more what his pupils could do. Today a great number of other school-teachers trained

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Part V Elementary Mathematics

in his approach do the same in many countries, and it is this, above all, that is so moving.

Thanks to the rods, charts and cards produced by Cuisenaire, we have given joy to thousands of children, and with this material they can do marvels. No longer being confined in a rigid and abstract world that has nothing to do with their mental life and their interests, the children now move in freedom in the relational world, and show us how much more capable, sure of themselves, and intellectually bold they are than any official number-psychologist ever told us they were. Teachers ought to know that their pupils, all of them, are much better mathematicians than the most visionary of educators ever ventured to say, that the whole of child psychology in the field of number has yet to be explored, and that the new methods I am recommending are so far-reaching that it would seem foolish to state their limits or to calculate how much a talented educator can do for the child.

Readers who are not familiar with Numbers in Color will no doubt be saying: “Whatever does this fanatical mystic mean about arithmetic and the child? Everyone knows that it’s torture for almost every child.� That is precisely the meaning of this anniversary message: that with Numbers in Color, the school life of the child is changed. Mentally, it is free, affectively, it is true. The child acts mathematically, he behaves as a mathematician in the way you and I do, if not more so.

Since he has only to perceive the relationships contained in his own field of consciousness, the pupil no longer finds difficulty in

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4 Arithmetic and the Child

expressing them, and, in doing so, he is expressing mathematics.

If he says 5 or 7 or 11 or 13 are such that they cannot be broken down into products because it is not possible to form them by placing rods of one color end to end (except for the white rod ( = 1) and the rod itself in the cases of 2, 3, 5 and 7), he has simultaneously satisfied a common property of these numbers and a criterion for determining whether a given number possesses that property. When he says that if the two terms of a fraction are doubled or tripled and so on an equivalent fraction is obtained, he shows in this way more than by writing the equation that he has understood the process of the formation of fractions as ordered pairs of numbers, and demonstrates what relationship he sees in the pair. When he arranges a set of rods in such a way that all the reds, all the greens, all the tans and so on, are together, and recognizes that any one of them adequately represents each of the others in the relationships which contain them, he has shown that the relationships of equivalence are directly available to him, and that for some properties (such as, for example, order) he can consider the quotient-set formed of any sub-set of ten typical rods.

We are now well advanced in the teaching (that is, in the becoming aware) of modern mathematics in the primary school.

But even more spectacular is the immediate dialogue into which the child enters with the material in order to find mathematical structures in it which permit a rapid and a correct algebraic dynamism. In fact, the characteristic feature of mathematicians is to work in the virtual, to perform only those actions that are indispensable, combining them mentally to transform them and save time, and to leave the others aside. This the child will do from the outset of his own accord and will do very well. Teachers 27

Part V Elementary Mathematics

must take advantage of this to allow the child to make swift and effective progress in his dialogue with relationships; those responsible for curricula must take it into account to avoid their confining the pupils in an a priori framework resulting from observing children traumatized and disheartened by an intellectual study which had no roots in them; parents and educational authorities should also know it so that children and teachers might be guaranteed the benefits of this work in the form of the joy which leads to mastery and competence and arouses new hope in those classed as handicapped in our society as a result of their arithmetical ineptitude.

I have worked with tens of thousands of children in very different surroundings from Glasgow and Dundee to Alexandria and Cairo, visiting Belgium, Holland, France, Germany, Spain, Switzerland, Italy and Greece, going anywhere and accepting any class, whether they were deaf-mutes, so-called backward children, super-gifted or normal, from five to eighteen years old, having perhaps a mastery of their languages, sometimes knowing only a few words: everywhere my findings have been the same, and each time I have obtained comparable results, and almost always, in the opinion of observers, spectacular.

This all proves that we do not at present expect of the child what he is really capable of giving, and that if we had adequate tools we should not expect to find a great difference between children considering a situation from the point of view of the relationships to be found in it; local language and culture are factors over and above the pupil’s mental activity— they do not cause it. Moreover, the information that the teacher possesses and sells is not the only channel of knowledge (indeed the 28

4 Arithmetic and the Child

opposite is true); the child can take the responsibility himself for his own education with teach-yourself methods, of which the Cuisenaire material is a striking example.

Today we can say to all school-teachers in the world: “If you devote your attention to the methods of school arithmetic we offer you today, your classrooms will be, both for yourself and for your pupils, lively and cheerful places where more profound and far-reaching work will be done to the satisfaction of all.�

29

5 Georges Cuisenaire’s Numbers In Color

This article first appeared in Belgium in Mathematica & Paedagogia, No. 4, 1954-1955.

There was once a schoolteacher who knew no more about mathematics than any other teacher, but he loved his pupils so much that he wondered what he could do to make the compulsory study of mathematics easy for them and a source of joy.

Where did he go to find the answer to his question? It was useless to ply mathematicians with queries. They do not understand the difficulties children meet. Nor did psychologists seem to be able to offer any more help since the knowledge gained by psychology of what the child can do can only with the greatest difficulty be separated from the educational system conditioning the pupil; moreover, the psychologist projects his own knowledge into the research being undertaken.

31

Part V Elementary Mathematics

Here, indeed, was virgin territory; a new idea to throw truly fresh light on the problem was needed. It was in music that this schoolteacher from Thuin, in Belgium, Georges Cuisenaire, a lifelong musician, found the idea. It is common knowledge that there are intervals between the notes of music, that certain combinations of notes are more easily retained than others, and that, in the various cultures of the world, once given a keyboard, a multitude of possible ways of combining notes and of objectifying the pieces have been found.

Why not give the child a keyboard on which he could recognize and find over and over again at will the most varied of abstract relationships? This was the new form that Cuisenaire gave to the problem of his research. Thus, with all his experience of children, he set himself to examine what the sensory motor world might offer to meet his first requirement. The answer, which was long in coming, was at last found: color. Color is a factor that is accessible to the minds of almost all humans; its shades and contrasts can act as a sign to substitute for the abstract notions that it is proposed to attain. Furthermore, as it is a sign and not a definitive static thing, we are here so close to the abstract that we may mistake the one for the other.

From this discovery Cuisenaire proceeded to create a large number of identical keyboards, made up, as on a piano, of a limited number of notes (ten in this case); and instead of sounds to produce, he gave them colors. His choice of colors was based on considerations that would require a full research program and we shall thus not deal with it here. One factor should, however, be observed: the first note is white, and goes into all other notes a whole number of times; the seventh is black to 32

5 Georges Cuisenaire’s Numbers In Color

underline its singularity in the series, while the three primary colors are used as the basic pigment to create three families whose relationships are obvious, red for the second, the fourth, and the eighth; blue for the third, the sixth, and the ninth; and yellow for the fifth and the tenth. In fact, the ten colors are all different and serve to distinguish rods of 1, 2, 3 . . . 10 centimeters in length respectively, all with a cross-section of one square centimeter.

Many of these keyboards together make up a set of rods, structured into sub-sets by the color and the corresponding length, making each rod of one color equivalent to all rods of that same color. Thus the relationship of equivalence is immediately accessible. Similarly, the set can be ordered, or well-ordered, or completely ordered, so that the structure of order is displayed at once in its different aspects. Moreover, it is enough simply to attempt to build any sort of construction with these rods—as all children do of their own accord as soon as they are given the material—for the possibility to appear of putting two rods end to end to obtain a new length; then the inverse operation can be introduced on the whole set, replacing two rods end to end with another rod from the set. This is what is known as an algebraic structure.

It is well known that one can do mathematics as soon as one becomes aware of these fundamental structures, and most of our mathematical thinking, which is not identified with arithmetic, corresponds in fact to the effect of one structure upon another.

33

Part V Elementary Mathematics

With the help of the Cuisenaire material this attitude of the modern mathematician can be brought to the kindergarten and to even younger children. It remains only for the teachers to maintain this attitude in all studies, making the way they teach ever more effective through this new awareness.

This Belgian apparatus, or similar material which is now being produced in other countries, has been submitted to tests for over 18 months, with thousands of children in various countries in hundreds of lessons observed by thousands of primary and secondary school teachers and students at all levels of education. It is eminently flexible and adaptable. Its richness is a perpetual source of amazement to those who do not know that the presence in the material of structures of algebra and order makes it isomorphic to a finite, yet indefinite, set of rational numbers.

Because the pupil is capable of a rational dialogue with the material he is acting mathematically from the very start. The only variable is the cardinal number of the set with which he is working, and, consequently, his findings too will depend on this variable only when it intervenes.

In this remark lies the germ of a revolution in the teaching of mathematics, and it would be of worth to reflect upon it.

The set will, in fact, enlarge proportionately as the child grows, but the operations will all be attempted on each set until the learner can take them as the object of his attention, and thus

34

5 Georges Cuisenaire’s Numbers In Color

leave arithmetic properly so called and embark upon abstract algebra.

Let us take an example to clarify the procedure.

We shall suppose that we are to make up the number 12 with two rods. We need an orange and a red, a blue and a green, a tan and a pink, a black and a yellow, two dark greens, and the same rods in reverse. With three rods, there are more combinations, and more still with four, five and over. In this way, all the partitions in whole numbers of 12 can be obtained using addition through the technique of placing rods end to end.

Subtraction thus becomes the realization that the situation can be reversed (the colors make this evident), so that 12 = 7 + 5 can be expressed as 12 – 7 = 5, or 12 – 5 = 7.

The same can be done for all similar situations, providing immense experience of the dynamics of addition and subtraction on the finite set of 12; and these situations are expressed verbally through correct statements about what happens within this set.

But, among these partitions, there are five formed of rods of the same color: those in dark-green, in green, in red, in pink and in white. These can be placed side by side and studied.

Instead of saying 12 = 6 + 6 = 3 + 3 + 3 + 3 = 4 + 4 + 4 = 2 + 2 + 2 + 2 + 2 + 2 = 1 + 1 + 1 + 1 . . . + 1, if it is realized that what distinguishes these written expressions is the fact that 12 can be 35

Part V Elementary Mathematics

said to be equivalent to 2 sixes, 4 threes and so on—new and remarkable decompositions of the group 12 can be found, engendering various ways of speaking of them. We can say, as above, 12 = 3 × 4 = 4 × 3 = 6 × 2 = 2 × 6 = 12 × 1. But we can also say that 3 is a of 12, 4 is a , 2 is a , and so on. Hence the fraction relationship appears through comparing one rod with another. Naturally, if 3 is a quarter of 12, one can find 2 quarters, 3 quarters, and 4 quarters of it, a third, two thirds and three thirds, and so on.

But, also, if we call factors of 12 the integers 1, 2, 3, 4, 6, 12 recognized as such above, it can be seen that they can be paired in certain ways to obtain the product 12. These operations go together: to say that 4 and 3 are factors of 12, or to say that 4 is a third of 12, or 3 a quarter of 12, are different awarenesses of the same situation, and should be recognized by this fact as being, in a certain way, equivalent. The general understanding of these equivalences should not be left until the child is ten years old; the understanding of sets of up to about 20 is certainly within his reach at the age of 6, and these should be made available to him.

Let us add in passing that ‘division with remainder’ comes in as naturally as fractions in the group of decompositions; indeed, it occurs more frequently than ‘exact division’. It is, in particular, always the case for prime numbers. Thus examples of these are met from the very outset. If we set out to make up a number with rods all of one color, we either find that this is possible, and one of the factors of the number is represented by that rod, or else a different rod must be used to complete the magnitude; this rod is always smaller than the other rod used and it can be 36

5 Georges Cuisenaire’s Numbers In Color

said that if r is the length of the remainder, and d that of the divisor, D=d×n+r

r<d

D being the original length in question, and n the number of divisor rods. The writing of this expression becomes a habit, so frequently does it appear; and it can be seen that, in the progression towards decompositions of ever larger numbers, the arithmetical processes that are contained in the situations are simply being recognized as algebraic operations.

The important lesson for the teacher to draw seems to be this: Monsieur Cuisenaire’s rods do not require that multiplication tables be studied before any study of division can be started.

Indeed, these tables never are studied—the learner knows them, having acquired them in their dynamism which gives the products, factors, and fractions of each number as it is met. It is obvious that the reverse of the product is the fraction, and that the academic language which speaks of exact division and division with remainder can be left completely aside.

To understand everything that is contained within a given situation is the true act of the investigator; pupils and teachers alike can now adopt this attitude to the benefit of all.

Another important lesson resulting from the structure of the rods relates to fractions and rational numbers. Each fraction can be seen as an ordered pair of rods, one serving to measure the

37

Part V Elementary Mathematics

other; thus each pair gives two fractions, one being the reverse of the other. But there is an infinity of fractions that are equivalent. Nevertheless, they are easily distinguished by their attributes of color or length. This class of equivalence which is expressed by the relationship that is present in all these fractions is the rational number which enters into the operations. Otherwise, no-one would be able to add

and

.

Addition can only be carried out if whole numbers are involved; hence of

+

becomes possible when, in the family of equivalence

and in that equivalent to

, fractions are found using the

same measuring unit. Then the two whole numbers are added together and a fraction is found which is part of a class of equivalence; from this class one fraction can be chosen which will be called â&#x20AC;&#x153;irreducibleâ&#x20AC;?.

This presentation of fractions as families of ordered pairs of integers rather than as parts of a whole seems well worthy of recommendation to teachers at all levels. But also fractions can be conceived of as operators and their algebra introduces a fraction whose effect would be equivalent to the succession of the effects of two fractions and is called addition or multiplication of these fractions according as the two operations act upon the same entity, or the second on the result of the effect of the first. By reversing the operations on operations, subtraction and quotient of two fractions are defined. With the

38

5 Georges Cuisenaire’s Numbers In Color

Cuisenaire rods, the teaching of this difficult part of arithmetic can be easily renewed.* Although we have already shown to what extent Monsieur Cuisenaire’s invention may be considered the most important contribution that we could have hoped for to solve our problems in teaching arithmetic, we shall now devote a few lines to its other applications. Combinatorial calculus becomes child’s play.

Theorems on the theory of groups of substitutions become obvious, and are evident even for 10 year old children.

Arithmetical and geometrical progressions can be more or less completely dealt with at the age of 10 and present a clear symbolism from the outset.

In considering the extremities of each rod as distinct, or else each rod as an ordered pair of points, we can introduce vectors on a straight line; and, with the observation that there is a symmetry with regard to an arbitrary origin on the straight line, the algebra of the set of positive and negative numbers can be made very clear. In particular, a modification of the rule of signs by the logical multiplication of two pairs of opposite transformations can be demonstrated and offered to the learners for reflection.

Some geometrical questions too can easily be put forward; for example, with rods of single type cubes can be formed. The pupils can then be made to find the number of rods necessary, to

* Cf. Numbers in Color, by G. Cuisenaire and C. Gattegno, Heinemann 1954.

39

Part V Elementary Mathematics

calculate the areas and volumes of the cubes, and, by this route, led to the statement relating to the relationship between areas and volumes (respectively) of similar prisms with sides a and b units.

Naturally, those whose task it is to teach other things than computations will want to know how these rods can be used for work on quantities. It is not difficult to see that relational mathematical thought is linked to nothing but itself, but when quantified magnitudes (such as weights or volumes) are being mentally considered, an adjustment of the mind to the chosen magnitude is made and a climate in the mind is created in which perception and action are mobilized in the direction of the experience recalled by this magnitude. Thus, in problems concerning money, temperature, or weight, the arithmetical processes are more or less the same, the only difference resulting from the fact that a temperature is not, in daily life, an additive quantity, that money (in Britain, for example) draws on almost arbitrary relationships, that weight is additive, and follows closely the pattern of numbers and so on. It seems that it is possible to learn number without necessarily referring to social experience, but that as life experience widens concrete meanings are added to the numerical one. Children find no difficulty in making this adaptation if their initiation is correctly begun.

Conclusion Something has changed since Numbers in Color made its appearance in the teaching world. We need not look any longer

40

5 Georges Cuisenaireâ&#x20AC;&#x2122;s Numbers In Color

for the way to improve teaching. This has been found by Cuisenaire. The joint efforts of a number of people have already worked out many uses for this tool (including the teaching of languages); and all that is asked of the primary or secondary school teacher is not to wait until everyone has profited from this gift before trying it themselves. When everybody has devoted himself to the techniques which allow these means to be perfected, we shall at last be able to say that the science of teaching and its technology have been founded to complete the art of the educator.

To offer everyone this complement to his pedagogical efforts is one of the joys that crown an intensive labor, and it is a joy we now experience.

The power of these colored rods lies, mathematically, in their penetration to the core of relationships and structures, and, psychologically, in their stimulus to intuition and enquiry.

Who can tell how much is contained in this new tool which is participating in a miracle?

41

6 The Study of Arithmetic with the Help of Color Associated with Length

This article first appeared in Le Courrier de la Recherche PĂŠdagogique, No. 4, Paris, January 1956.

Quite a number of teachers systematically use color in creating educational material. The problem I am tackling here is more special. I want to give an idea of what I have found in my own investigations which have been conducted for over two years with thousands of children in several countries using the material devised by M. G. Cuisenaire of Thuin, Belgium with the purpose of simplifying the transition from concrete to abstract in the teaching of arithmetic in primary schools.

Here the problem is considered from the mathematical, psychological and pedagogical points of view, sufficiently clearly, even though briefly, I hope, to allow the interested reader to take it up in a critical and constructive manner.

43

Part V Elementary Mathematics

1 In studying the childâ&#x20AC;&#x2122;s conception of number, psychologists make use of a point of view which results from an adult analysis of the idea of number. Clearly many interesting things may be said from this point of view. The books of M. Piaget bear witness to this. But it would be ridiculous to believe that there is no more to be discovered in this field, and I wish to show here that a new point of view now exists which could not have been foreseen either by mathematicians or psychologists.

Since there is room for genius in all human affairs, questions renew themselves as soon as a truly original point of view appears.

Georges Cuisenaire has created a very simple educational material consisting of colored rodsâ&#x20AC; in which color is used to convey size and various relationships. But a given length is not rigidly associated with one color only, since this length can be produced in many ways by different grouping of rods of various colors.

When active use of the rods has been established, color is maintained as a sign for a certain length, the rods being taken away and hence completely representational work, mathematical in character, can be undertaken. This is the point which was not foreseen by the a priori school of mathematicians nor by the logicians.

â&#x20AC; The description of which is omitted here.

44

6 The Study of Arithmetic with the Help of Color Associated with Length

In fact the question facing us was this: when can it be said that number exists in the mind of the child?

However, mathematical reality is more complex, and the study of what the history of mathematics has to offer rapidly assures us that number, even the integer, has taken a long time to emerge and be defined. To require that it be found in the childâ&#x20AC;&#x2122;s mind, conforming to its pure definition, in my opinion amounts to the imposition of a strait-jacket on research. What is closer to the way in which the human mind has dealt with these questions in the course of its mathematical dialogue is to perceive the releasing, the freeing of thought engaged in certain activities. If it is true that it was necessary to wait until the end of the last century for mathematicians to break free from the hold number had on them and to direct themselves towards the analysis of qualitative mathematics, then, at the same time, the study of the problems involved in the apprenticeship to arithmetic cannot fail to require the consideration of a variety of points of view.

Nowadays we have learnt to look at situations with more perceptiveness and in a more tolerant way and to let lessons evolve rather than ignore them to substantiate a way of seeing things.

If we therefore set ourselves the task of understanding what really takes place in the mind of the child learning arithmetic, we must be able to protect the child from our interference and put him face to face with the object of his dialogue in order to be able to observe him. Here, observation must not be taken to be the mere recording of his reactions but must be a genuine

45

Part V Elementary Mathematics

attempt to understand the steps taken by the child in his selfeducation. In fact, it is not possible to define these steps except within and with respect to the situations in which the child is found to be placed.

In giving the Cuisenaire material to children, I have had the opportunity of observing, among many other things, that most can in a very short time and of their own accord master the algebra contained in simple exercises set them, regardless of local climatic conditions or of any other influences to which they may have been subjected; moreover they are so aware of it that they express themselves, whatever their language, in a mathematical way. Children of five are mathematicians if the sets which they have to structure have few elements (say 10); at six and seven years of age they still act mathematically even if the sets are enlarged to contain up to or more than 100 elements.

Not only do they spontaneously discover the obvious structures embodied in the material by the inventor but also those of which he was unaware. In fact, if a child is allowed to play with the colored rods he soon recognizes three fundamental multivalent structures of modern mathematics: â&#x20AC;˘

The relationship of equivalence, that is to say, rods of the same color are the same length (owing to the way they are made), and those of different colors are of different lengths. Moreover, what is known in mathematics as the quotient-set is also noticed when the child represents with any one rod of one color the class of equivalence of all the

46

6 The Study of Arithmetic with the Help of Color Associated with Length

rods of that color. Clearly, the still more general notions of set and sub-set have been grasped even if they have not been formulated.

•

The relationships of order, that is to say the fact that, taking two rods a and b at random in the set, the child can say if a equals b or if it differs from b, and when he agrees to describe a as smaller than b (or b larger than a) or vice versa, he links this word to the perception of inequality. Such a comparison between rods is more profoundly structured when the child can combine pairs of inequalities to form a transitive set of propositions: if a < b and b < c then a < c, and the like. Not only is the given set of rods ordered, but likewise every sub-set of this set.

•

The algebraic relationships which result from the introduction of an operation on the set of rods.

Every child spontaneously arranges his rods in various ways to produce an extraordinary variety of color patterns. If he becomes aware that two rods end to end replace another rod, or two other rods end to end, in length, he explicitly introduces an algebra into the set: an algebra represented by one operation marked by the sign + and called addition such that its obvious properties are a + b = b + a, (a + b) + c = a + (b + c), (commutativity and associativity). From a + b = c through a new awareness (and a new notation), a = c – b, or b = c – a are drawn. Subtraction is introduced as the inverse operation of addition, and addition appears as the inverse operation of subtraction. If a second operation is introduced and called multiplication, distributive with respect to addition, the algebra is that of a commutative ring; when the inverse of multiplication and a new 47

Part V Elementary Mathematics

neutral element are introduced, this transforms the structured set into a field. Multiplication can easily be seen as the iteration (repeated addition) of one and the same rod.

As the child can achieve all these fundamental structures when he is given a set of rods, and also since he can recombine them to obtain more special, fertile and rich situations, his mathematical self-education can take place on a realistic basis which conforms with the working of the mind.

And it is just because the qualitative can be mathematised and is more accessible to the child that modern mathematics is nearer to the childâ&#x20AC;&#x2122;s mind than is the mathematics we try to force on him too soon.

2 For the moment we have only observed the beginner in his first contact with the Cuisenaire material. But the child grows and his mind combines the forms and ideas which he absorbs in his dialogue with the material. He grasps, for example, that the same situation is susceptible of several interpretations and that mental dynamism transforms it to the point where it becomes hardly recognizable.

Thus, in these actions which are always the same and which have enabled him to form different equivalent lengths with the rods, he discovers that some of them can never be formed by iteration of rods or by combinations of rods; in other words, if he does succeed in forming the lengths in this way, the composite numbers (and therefore their factors), are found, and when he does not, the prime numbers. However, from the point

48

6 The Study of Arithmetic with the Help of Color Associated with Length

of view of addition, and even of division with remainder, the two types of lengths do not appear distinguishable from each other.

The small child will not arrive at the most general statements possible, but in relation to the numbers which he explores to expect this of him at a time when one is concerned with determining by means of experiments whether he understands number or not is to demand what is illegitimate. The child may have found the means of deciding these questions but it will be insofar as his number experience will permit it.

Moreover, it is easy to see that for every mind, young or old, mathematical or otherwise, there are three levels with reference to numbers. There are familiar numbers about which many things are known and whose number varies considerably from one mathematician to another. There are those numbers about which one has some idea and which can be studied more closely with a certain amount of difficulty, and there are the numbers which are never considered at all but which one feels must exist, those requiring more than, let us say, a thousand digits to be written.

Why, then, should we not keep up this situation at the pupilâ&#x20AC;&#x2122;s level and make a thorough study of useful numbers, take a glance at numbers which may be interesting, and leave the others alone? The dividing line between two contiguous classes will shift with growing experience.

Let us add that the mathematicianâ&#x20AC;&#x2122;s true activity consists in considering that the virtually possible is directly possible and

49

Part V Elementary Mathematics

realizable, and that, so long as there is no reason why he should stop thinking that some process, which had been found to work in the number of cases considered, should work, he will consider this process as if it would always work. For example, when he will find the lowest common multiple or the highest common denominator of two small numbers, and when he will have seen that this operation in this case can be expressed in a manner independent of the case, he will have a procedure on which to work in order to determine its properties. The mathematicianâ&#x20AC;&#x2122;s most common activity is of this kind, and it has little to do with logical reasoning.

The child can grasp such mathematics though he may appear to be totally incapable of following a logical line because he is little interested in it.

It seems proven to me by all the work with children whom I have confronted with the Cuisenaire material that, if we approach them as mathematicians, they respond very well, and if we approach them as logicians they leave us high and dry with our ridiculous questions. Their knowledge of some numbers is remarkable though their understanding of â&#x20AC;&#x153;numberâ&#x20AC;? itself can be very limited.

3 An actual experiment. Here, as an example, is a lesson given on 25th May, 1955, at Zurich to a class of 8 year old children who had had one year of city primary school; they started school at the age of seven and the school year there begins at Easter.

50

6 The Study of Arithmetic with the Help of Color Associated with Length

These pupils, both girls and boys, had studied the numbers up to 20 during their preceding year and, between Easter 1955 and March 1956, would have to study those up to 100. I had no specific assignment for this lesson and the pupils had been using the rods for the study of addition and subtraction since February. So I thought of investigating the obstacles arising in the study of certain fractions and which were sure to have been left untouched at the school as they were not included in the local syllabus for children under the age of ten.

Having established that the pupils without difficulty added any rod to any other simply by feeling them behind their backs, and seeing that a firm association existed between the first whole numbers and the length of the rods recognized by their color, I began by asking what 2 Ă&#x2014; 7 is. Then, having received a correct answer from all the children, I again took the black rod (7 centimeters) and asked what its double and quadruple was. We put four rods on the table and measured, with the help of the orange rods, that the answer was 28. I asked what half of 28 was, then a quarter.

Then we formed 28 with the help of the 4 centimeter rods (pink) and saw that seven of them were needed. I asked again for a quarter of 28 and for them to show me the rod which represented it. Everyone, without exception, held up the black rod. Then I dared ask the question: Show me of 28, after having obtained the answer that seven pink rods were needed to make 28. Nobody knew the answer. I asked the two questions in the above order several times, and one pupil showed me the pink rod as the answer of of 28. There was obviously a language difficulty, and I gave the definition of

51

by saying that

Part V Elementary Mathematics

we call one seventh of a length (in this case 28) the length which goes into it 7 times. Then three pupils showed me the pink rod.

The question was put again, and care was taken to ask those who had found the answer to explain what they had found; the number of those who discovered the answer by themselves increased. But there were still some who replied that a seventh of 28 was 28 or 7. To satisfy myself that there was no imitation among those who had found the answer I asked to be shown of 28, but this time in a single rod. In a few moments a good number of tan rods (8 centimeters) were held up.

Having failed in my efforts to make certain children undertake the mental search for a solution to a problem quite new to them, I tried the following way. Cuisenaire had introduced the notation of two rods placed crosswise one over the other to show their product, so I first tried to obtain the reading of a product through the factors; for example, instead of saying 28 the children were to say 4 × 7, and then I should reverse the order to 7 × 4. We first attempted to read known products: 2 × 3, 7 × 2, 5 × 10, and so on. We read them all in both ways and gave the product when it was known, but when it was not the children were always able to say 7 × 8, 9 × 7, 9 × 6, and so on, without giving the result. This is a case of a simple mental conditioning and verbal association.

In certain cases there is a further operation which would provide the known answer, the name of the result. I made no effort to make them learn the names of the products, all I was looking for was the mechanism whereby two steps which provide fractions of a number represented by the product of its factors, could be associated with a sign which, it was realized, represented a multiplicative association of integers. Thus, if 4 × 7 is the 52

6 The Study of Arithmetic with the Help of Color Associated with Length

product, it can be seen that a quarter of this product is 7 and that a seventh is 4, without knowing the relationship 4 × 7 = 28.

In order to make sure that the mental activity had passed from an understanding of addition facts, which the rods placed end to end gave us at the beginning, to an understanding of multiplicative bonds, I took the “table of products” invented by Cuisenaire. This table is organized according to the following principles: an arbitrary design, composed of a white circle between two crescent shapes colored according to the color code used for the rods, represents the factors (up to, and including, 10) of the product. The white circle can be covered by a counter bearing a numeral. For example, 15 would be placed on the white circle described above where one crescent shape is green and the other yellow (representing 5 × 3); the symbol to represent 16 = 8 × 2 = 4 × 4 would be two pairs of crescent shapes grouped around the white circle in the form of a cross, the members of one pair being colored red and tan and those of the other both pink.

At once the products 4, 6, 8, 10, 12, 15, 25, 50 were recognized.

Without going any further I showed them that if I said, for example, one fifth of a number which I did not name but which I pointed to with my finger and where yellow appeared, they were immediately to give the other number appearing in the symbol.

In this way they recognized of 12 to be 3, of 25 to be 5, and of 50 to be 5.

In fact, we were confronted with the following mechanism: calling c the unknown result of the product a × b, we could with

53

Part V Elementary Mathematics

certainty tell that a = the ath part of c.

, or that a = the bth part of c or that b =

In a few moments the pupils could answer correctly that a ninth of 9 Ă&#x2014; 6 was 6 and a sixth was 9, and none of the products presented in this way offered more difficulty than any other.

If these pupils learn the value of the products, the mechanism described will remain exactly the same, but here one or two sentences might be added. For example, it may be asked what a certain product is, say 56, and what or of it is; the answer will be given immediately as 8 or 7, obtained as before from the table. During this lesson the mechanism was broken down and so taught me about the quick work done by Cuisenaire pupils who seem to be prodigies. In fact this part of the lesson, though it surprised some experienced teachers who were present, was remarkably helpful towards making me understand the way in which the mind of the child works when learning arithmetic with the Cuisenaire material. The childâ&#x20AC;&#x2122;s method is the same as that of the mathematician who, having been able to abstract an operation from a multitude of actions, through this operation, can, in a few moments undertake virtually a sequence of mental steps which, had they been actually performed, would have demanded great effort. Mathematics training consists in making everyone expert in the art of avoiding useless steps and of applying a mechanism of compensation which facilitates his task. There was no effective understanding of what or meant because the rods had not illustrated that in fact 7 or 9 rods of a certain color were necessary to make up a specific length. But there had been a purely mental operation and

54

6 The Study of Arithmetic with the Help of Color Associated with Length

adequate verbalization, and a small margin of uncertainty still existed which one or two lessons would have sufficed to eliminate.

To end the lesson I returned to the top line of the table of products formed by the products, 4, 8, 16, 32, and 64, each, with the exception of 4, being the double of the preceding one. On this line we could double numbers they knew, or take half of them, but we could also iterate the operation and obtain of 32 or of 64. And, once this had been thoroughly done, we doubled 64 and even 128. The delight of the children, when they heard that after 128 they also had to double it too, was a clear sign that this incursion into the field of larger numbers held a great attraction for their minds.

4 Through this method we are far removed from the customary precautionary exercises where the addition of 1 is the measure of everything, and though it may be difficult completely to justify a lesson like the one above which crammed so much into three quarters of an hour, I do not hesitate to say that the results of similar experiments in hundreds and hundreds of classes with several thousand children of all ages and levels confront us squarely with new psychological and pedagogical problems.

Up to this point our pupils have been paralyzed by our logical conception of mathematics, and we ourselves, petrified by the mathematiciansâ&#x20AC;&#x2122; demands of rigor, have tried to get from our pupils, prior to mathematical experience, its codification.

If we allow intuition to play its part, if we do not require that our children be pre-logical in the logical sphere, but if we allow them 55

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to organize their own experience in accordance with the criteria of which they are capable within sets they can dominate, the result will resemble what the mathematician offers in his creative field.

The social factor has been over-stressed in the childâ&#x20AC;&#x2122;s apprenticeship; we have been too concerned with the manner in which he absorbs adult knowledge to the point that some find it shocking that the child should know anything which adults have not taught him. However, anyone can satisfy himself that the child, if placed in a situation commensurate with his ability, upsets all our preconceived ideas of his reactions and offers an original point of view to the keen observer.

I do not think that I am wrong in stating that the introduction of color as was conceived by Cuisenaire to mathematics teaching, has made it possible to free the child from a thousand obstacles put in his path by the blundering adult. Our learned psychologists will have much pleasure in revising their studies of the childâ&#x20AC;&#x2122;s numerical thinking if they forego questioning him with a view to finding out at what point he thinks like certain adults, but instead, use for their edification a complex situation where the child is absorbed and creative, as in those described above.

Teachers too will find in the Cuisenaire material a basic means to renew their teaching which for centuries has been kept in the same arid state through the over-emphasis of the unit and the absence of true contact with the childâ&#x20AC;&#x2122;s enquiring mind, which is much closer to the conception made available by modern qualitative mathematics.

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7 Learning Mathematicsâ&#x20AC;&#x201D; A Practical Solution

This article first appeared in Impulse, No. 2, September 1957.

After centuries of mathematics teaching reserved for a small minority of the inhabitants of a few countries, we are today confronted with the challenging problem of how to equip vast numbers of people with the efficient use and interpretation of mathematics.

While the usual approach to the problem is to suggest watering down the grammar-school syllabusâ&#x20AC;&#x201D;thereby producing individuals with neither experience of mathematics nor a sense of its valueâ&#x20AC;&#x201D;it is my experience that it is possible to ensure deeper understanding and greater all-round efficiency by the reverse process: by broadening the syllabus and including in it much more recent concepts and techniques.

There is no question of suggesting that people can be made more intelligent. It is a matter of using the human intelligence to 57

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far greater effect which has been made possible as a result of two discoveries, the one psychological and the other methodological.

Substitutes for Action The first can be expressed as follows: the path taken by thought in reaching mathematical awareness is through substitution of virtual for actual actions, and the extension of the field in which the virtual actions are operative to its maximum. Let us consider, for example, the various movements of our bodies. We can spin, we can walk upright, we can bend our backs and our knees, and we can compose certain of these actions. Once we have actually done one or more of them, we have memory of them and they can be evoked. In sliding along the floor in an upright position, for example, we find that each part of our body remains at a constant distance from the floor and that we describe in space figures of which we have some mental conception. Some of these can be schematized in drawings which are symbols substituted for the actual happenings. In talking about them we find we are aware there is always a reality supporting the words. It is a dynamic reality, and if in the learning process this dynamic reality is constantly present, there is no difficulty in maintaining understanding throughout.

The reduction of mathematical understanding to an awareness of virtual actions is indeed the key that can open mathematics to all, but there still remains the problem of finding the techniques to serve each purpose. It is proposed in this article to give some idea of one of the most important of these, which has solved the

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problems of teaching arithmetic. This is what was referred to above as the methodological discovery.

A priori it would seem absurd to mathematicians to base the concept of number on anything but counting. It has been universally believed, in spite of the difficulties encountered by logicians, that the surest definition of numbers was by means of the operation of adding the unit to itself, and age-old methods of teaching arithmetic are based on this belief. The variety of materials given to children in infant schools to provide experience of numbers are all essentially based on counting.

They are made attractive, for aesthetic and commercial reasons, and color appears as an added element. One man, however, whose thought went deeper, has produced, contrary to all opinion, a different foundation for number in which color is essential and no longer incidental.

Color and Number This man, Georges Cuisenaire, a Belgian primary school teacher, geared his thought to producing a sensory-motor apparatus in which number relationships were readily accessible. Being a musician, he developed his material as if it were a series of notes on a keyboard. By agreeing to call C all the notes whose vibrations are integral multiples and sub-multiples of a particular note called C, and representing the notes of the scale by different values, a keyboard is possible on which all the tunes involving these intervals can be played. Cuisenaire’s ‘keyboard’ is based on colored rods. Those 2, 4 and 8 centimeters long have a red pigment (a major chord), 3, 6, and 9 centimeters are blue

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(a minor chord), 5 and 10 centimeters are yellow, 1 centimeter is white and 7 centimeters black. As the particular length increases, so the color deepens. Rods of 1 square centimeter section proved a convenient size for concretizing the material.

The colors finally decided on are the result of over 20 years of experimentation, and are very attractive when the rods are clean.

This solution to the difficult problem of eliminating counting as the basis of mathematical materials has brought with it an enormous improvement in the teaching of arithmetic as a whole.

The fact that the relationships selected for the various colorings relate lengths with each other and not only with the smallest one, which is white, has allowed us to give a place to qualitative arithmetic and thus to deal with the formation of arithmetical ideas at a level where numbers and their names are not yet necessary. Actual manipulation of the Cuisenaire rods gives rise to the virtual actions which lead to understanding, and through play with them we can take the pupil to the point at which a precipitation easily occurs when the signs 1, 2, . . . 10 and +, â&#x20AC;&#x201C;, Ă&#x2014;, á are introduced.

This will be easily understood by reference to the illustration* which shows a pattern with the orange rod at the top. Children have selected their rods so that at first two of them are sufficient to make the length of the orange, and then in the last two rows four and three are used. These actions are independent of the

* The six illustrations to this article, taken from the filmstrip on Numbers in Color, are omitted here.

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leading rod (here the orange) and will be the same whatever the length proposed. In this pattern, as can be seen at once, addition, subtraction, multiplication and division are contained.

Here then is the first obvious advantage: instead of studying each arithmetical operation in turn, children study all the operations simultaneously, but within sets which they can fully grasp. It is remarkable to see how this apparent complication of the situation proves in the event to be the simplifier of the learning process. Since a start can be made with quite small rods and limited numbers of decompositions of any length, children can fully master the actions and their virtuality, and as they move on to longer and more complex patterns they gain a new insight: that more can be said and more properties found as the numbers get bigger. Experience shows that children of seven are quite capable of arithmetic which seems difficult to many children of eleven using traditional methods.

Fractions and Classes of Equivalence Coming back to the material, we find that by making use of the fact that the difference between any two successive rods is the white one, we can meet all the properties of whole numbers, while by taking any group of rods related by color we can obtain relationships that lead to simple fractions. From the set of rods it can be seen at once that fractions are only a new awareness of what was already known. Thus, if the various rods are twice, three times, four times . . . ten times the white one, the latter is half, or a third, or a quarter . . . or one-tenth of each of the others. From this observation we see that there is another revolution on the way: the reading of a multiplicative relation (6 = 3 × 2) in the opposite way yields the fraction it contains (2 = × 6 or 3 =

× 6).

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Fractions are mental structures extracted from straightforward, simple situations involving the rods, and made evident both through the colors and the lengths. But still more important is the fact that instead of considering one fraction alone, we can now consider those equivalent to it as well.

The appearance of classes of equivalence in early arithmetic has completely altered the childrenâ&#x20AC;&#x2122;s experience and led them to behave as mathematicians. Anyone who has seen children at work with the Cuisenaire material, or studied the sequence of questions in arithmetic books based on this system, cannot fail to see that this is so. We have erred for a long time in giving children the wrong experience. It is now possible both to increase the syllabus considerably and to ensure mastery of it by ordinary pupils, and these two developments will amaze only those who believe that children who failed by traditional methods did so through their own fault. It is, on the contrary, natural for the child who masters a skill to advance more rapidly and to enjoy his conquest. The Cuisenaire material has succeeded in removing deeply ingrained prejudices, and many hundreds of ordinary teachers are showing every day that research and practice have followed the wrong path for too long.

Naturalness of Algebra The use of this material has taught us other positive things which are in line with discoveries made at the secondary school level a few years before the colored rods came our way. The naturalness of algebra, for example, can be seen when we recognize that it is essentially a dynamics of operations, that is,

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that the elements of the set on which we work are operations and that we operate on them. Thus we square sums, or cube a difference, or consider the equivalence of systems of equations one of which separates the variables. This awareness has made it possible to design a syllabus which I have called â&#x20AC;&#x2DC;the functional syllabusâ&#x20AC;&#x2122; (1949) and which takes pupils from the infant to the university level, providing each level (not age) with the situations and the exercises necessary to maintain full understanding and follow a line of development that opens new fields and forges new and more powerful tools.

Mathematics is essentially a saver of action precisely because, in the realm of the virtual, it is possible to see which actions can be combined and which replaced by a smaller number without actually trying it out. If the pupil lacks this awareness, mathematics soon becomes a tedious and uninteresting subject.

When he has it, the study develops, like spontaneous play, into a means of increasing the power of the mind. This awareness can be reached by pupils in all our schools, not through hearing the teacher talk, but by being confronted with the many learning aids which have been developed. In Britain, the Association for Teaching Aids in Mathematics has contributed to the propagation of these ideas by organizing, in conjunction with Institutes of Education, courses in which large numbers of teachers have taken part.

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8 Theoretical Remarks on the Cuisenaire Material

This article first appeared in Mathematica & Paedagogia, No. 9, 1955â&#x20AC;&#x201C;1956.

Those who have watched pupils using the Cuisenaire material can be in no doubt that it raises questions which go far beyond the field of primary education. Having used the material with so many children of all ages, over a period of more than four years, I have been constantly concerned with these questions, and my findings seem to me to be of general interest. I propose here to deal with two questions in particular: the nature of mathematical activity, and the basis of the formation of arithmetical experience.

The Nature of Mathematical Activity In speaking of mathematics we soon become aware that we have only a vague idea of what mathematics is. Our minds are indeed polarized in a definite direction, but it is not easy to find a 65

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satisfying definition of the vast edifice laboriously constructed over the centuries. We tend to forget that mathematics is not one but many, and that in its long history its nature has undergone profound changes. There seems to be no single definition which exactly covers what mathematicians refer to when they speak of mathematics. This being so, I have attempted to examine the activity of the mathematician rather than the edifice, and the results of my examination enable me to make what seems to me to be a precise and important contribution to the problem.

To put it briefly, the essence of the mental attitude which characterizes the mathematician is the substitution of the virtual for the actual, the performing of virtual actions which are envisaged as being real or realizable, once there is no reason to suppose that they might not become actual.

There is thus a series of awarenesses which are organized into a special power, into a set of habits, and which simplify complex virtual operations which are inserted into actions, the economic hierarchy of which becomes an ideal.

Any question which involves indefinite classes of objects or actions can serve as example. It is impossible actually to embrace all the positions of an object, but the mathematician finds it legitimate to make statements about them. The mental process which makes this possible is clearly the following: there is first the recognition that “position” signifies something which presents no difficulty to the mind, that “all” represents the set of which certain elements only are perceived, and that a statement

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referring to “all positions” is a statement about what takes place in certain positions, and at the same time a statement that there is no reason to suppose that these positions are privileged positions with respect to any others of the set. This process leads to an awareness of relationships, and of the virtual actions that underlie the relationships.

To my mind it is precisely this that constitutes the evolution of mathematical thought in the learner. There is no question for him of ideal entities, the perfect counterparts of imperfect physical objects. The mathematician does not operate on entities, in spite of our current use of the term. An entity, if such exists, is a moment in the awareness of relationships, held in the mind by an act of volition on the part of the mathematician, but capable of being taken up again and put back into the casting machine, which is a further act of awareness of what links this entity with the others. Its quality of being then disappears, is transmuted, and presents a new field for mental contemplation.

Is this not in fact the case with numbers, which have for so long been considered to be mathematical entities par excellence ?

Whatever the point of departure chosen for “logicising” the set of integers, it remains true that action is entailed: counting, merging together the enumerated elements, substituting a name for the set thus obtained, attributing to this name qualities which are no more than virtual actions which become increasingly complex and numerous, which in fact are unrealizable, and so on. What is important to notice is that once the counting stage is past, the unit that has served in it is

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ignored, and in fact, since actual actions are only possible in relation to real objects, it is only virtually that the indefinite addition of the unit is possible. We need to distinguish two procedures and two awarenesses: the one relating to the operation which consists of virtual addition of the unit, and the other to the awareness that the way lies open for this operation to be continued indefinitely.

This ignoring of the unit entails a property of whole numbers which is purely formal, the awareness that addition yields different whole numbers, different precisely because we register in our minds, in psychological time, that something has happened through the virtual action that we envisaged.

And what of the combination of units to form whole numbers? It cannot be a “new” whole number that is formed, since its predecessor disappears as soon as a further unit is added to it unless there is an inexhaustible stock of units, and once a whole number has been produced and labeled, we begin again, register mentally the earlier point of arrival, and add still another unit. It is because the living of these virtual actions gives rise to a mental stock endowed with its own dynamism, and to mental “halts” which objectify these actions, that we have at the same time an awareness of mathematical beings and of the thought which sets them going and transforms them. For the mathematician these beings have reality because he bestows on them the character of objects, but of mental objects still showing the mechanism that served to create them. But his activity does not stop there.

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Virtual actions, of whatever kind, are susceptible of being perceived in their dynamism against the background of any mental activity, and if the mathematician, instead of conceiving of beings and their propertiesâ&#x20AC;&#x201D;which are revealed by objectifying certain stages in his activityâ&#x20AC;&#x201D;examines the dynamism itself, he finds that his awareness does not exhaust what is possible or permissible within the framework of the virtual actions in question.

Thus going up a staircase is a real action, while to imagine oneself going up is virtual; merely to perceive the action of going up and the equality of the steps is to have a certain awareness of progression (which we call arithmetical), but to conceive that when the action is virtual neither the value of the step, nor the direction in which one goes, nor the number of steps is relevant, is to deprive the virtual action of all its links with the real action, but it is at the same time to recognize its virtual nature. When it is the dynamism which is being considered, a real action becomes a mathematical theory which provides moments of awareness which we call theorems. It is enough, for instance, to imagine the relationship of a given step to all the others, or of one step to any other, to grasp the mechanism of the mental halt and to contrast it with the dynamism. The latter yields, at certain moments, certain invariants which may be of great practical interest; at other moments it pursues its function as the creator of new situations.

Having given the example of the staircase in order to show how a mathematical theory develops out of an act of awareness of what in a habitual action is variable and what invariant, I should

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perhaps add some remarks to make these questions less abstract.

A geometrical progression requires quite a different model from the staircase. To consider arithmetical and geometrical progressions separately is to study mathematical relationships in which some awarenesses are fixed; whereas to leave aside the distinctiveness in these examples in order to conceive of what belongs to the idea of sequence and series is to return to the dynamism supporting the progressions and to leave the field free for it. If we find anything to say in this sphere, it applies to these series as much as to any other one. It is not a case of generalization but of stripping the dynamism from the remnants of actual actions. The return to dynamism allows us to transcend questions where structure dominates, that is, where the action can still be discerned.

The history of mathematics abounds in examples which could serve as illustrations, but we must confine ourselves here to seeing how this understanding of mathematical activity applies to arithmetical activity.

The Formation of Arithmetical Activity I shall restrict myself to a consideration of this activity as revealed through contact with the Cuisenaire material.

I have given the colored rods to children of all ages and have closely observed the apprenticeship to arithmetic by normal children, by deaf children and by blind children.

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It is obvious that with the aid of the rods there is, for the child, a long phase of qualitative arithmetic which provides all the mental structures and the dynamism which will form the basis of numerical arithmetic. Moreover, the latter can make its appearance much sooner than is generally supposed, and can be of a much more advanced level, as I shall presently attempt to show.

Manipulation of the Cuisenaire material is based essentially on two actions, that of placing rods end to end and that of placing them side by side. The rest is mental. It is therefore in the childâ&#x20AC;&#x2122;s becoming aware of what he perceives that we must find all the dynamism which is to give rise to the arithmetical mental structures.

The real actions can be performed, repeated, suspended, and the pattern produced can be destroyed, unmade, remade, imagined, reimagined, viewed differently, and so on.

The second awareness will be the awareness that rods can be placed end to end as many times as one wishes, that there is no theoretical obstacle to the repetition of the action. In fact, when the repetition is actual the awareness is much more complex, because each time rods are taken and placed end to end, their length and color will be associated with the action.

If the child is left free to play at will, he will become aware of degrees of freedom entailed in his action but also of their limitations. He will find that he can put rods side by side, one on top of the other, with this or that face upwards, at the same time

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as end to end. He will discover that certain spaces can be filled by some rods but not by others, and so on. With this experience he will become aware of what his action can, and cannot, achieve; he will learn to use his tactile sensitivity to reduce the number of his actions, and to avoid those which are useless or dangerous for his task; he will come to use his vision to form criteria which will precede action.

But all this will not yet be directed towards the mental polarization which leads to arithmetical mastery. This he will obtain by selecting his actions with a view to essentially operational acts of awareness by putting rods end to end in order to form a given length.

Beginning with a set of three rods, such that two of them end to end form a length equal to that of the third, it is easy to ensure that the pattern of the colors becomes the centre of attention, by first showing the three rods in the position we have indicated above, removing each one in turn and requiring the missing one to be found by the children. Very soon, trial and error will be replaced by the recognition that the colors are adequate clues for the solution of the problem.* It is obvious that here we have four qualitative operations of the type: a + b = c, b + a = c, c â&#x20AC;&#x201C; a = b, c â&#x20AC;&#x201C; b = a.

* This will not, of course, apply to blind children.

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Since there is a wide choice of such triplets, the opportunity exists for considerable experience through which the addition of a complement to one length in order to form another, as well as their obvious properties, can be firmly established.

This can then be done with four rods, such that three of them end to end make the length of the fourth, and all the operational possibilities, including permutations, are obtained. If the four rods are such that two pairs end to end form the same length, we find other results: a + b = c + d, b + a = c + d, a + b = d + c, b + a = d + c, etc. where the two contiguous rows play the same role.

The table of decompositions of a given length will contain a series of questions of a qualitative nature. For instance, the difference between lengths which we call odd and even will be seen from the fact that a white rod is always found in some rows, where the other rods of the same color go in pairs, and never in others; prime and composite numbers will be recognized by the fact that it is or is not possible to form rows all of one color (apart from the white).

We have called all this qualitative arithmetic because it involves only those properties of the Cuisenaire material which depend on the color and the relations of equivalence (rods of the same color are of the same length and those of different colors are of different lengths), and the action of internal composition by which two rods are replaced by one whose length equals the sum of the two others.

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Children who have become aware of these structures are ready to go straight into the study of arithmetic proper. This will be the set of virtual actions that result from the recognition: 1 that if the white rod is taken as the unit, the other rods will be equivalent to 2, 3, . . . 10 units respectively, and that the succession of numbers, as names, corresponds to lengths obtained by putting rods end to end one after the other in such a way that each time the length is greater than the preceding length by the length of the white rod; 2 that whatever the length chosen, the set of decompositions can be interpreted as additions, subtractions, multiplications, divisions with remainder, factors, fractions.

Qualitative arithmetic, then, provides the mathematical mental structures, whereas the decompositions provide the numerical habits, the experiences which are ordinarily classed as arithmetical. From the point of view of truly mathematical activity it matters little whether the result of an addition is 7 or 83; what matters is that the child shall be aware that if he makes an addition he can unmake it by the appropriate subtraction, that the evocation of factors entails that of the result of their multiplication, and so on. In fact, although what is always stressed is the acquisition of details, of â&#x20AC;&#x153;numerical factsâ&#x20AC;?, what forms arithmetical ability is mastery of operations. With the Cuisenaire material it is this mastery that is acquired, naturally and from the start.

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It follows that we are in a position to review the question of the syllabus for primary school arithmetic. From what we have learnt from our psychological experiments with thousands of children of many different backgrounds (differing in language, culture, political system, religion, curriculum, educational system, and so on), the Cuisenaire material clearly makes a reconsideration of the syllabus possible.

In the first place it should be pointed out that certain local practices, which seemed justified, are now seen to be based on mere prejudice. In Great Britain, arithmetic is begun at the age of five, in France and Belgium at six, in Zurich and Yugoslavia at seven, and so on. In France, at the age of seven, pupils must have studied whole numbers up to 100, in Belgium whole numbers up to 20, while in England there is no limit set and in Yugoslavia whole numbers are not studied at all in the first year.

In Germany and Italy, children are not allowed to study fractions before the age of twelve, but they must know all the properties of whole numbers up to 1,000,000.

What are we to make of all this variety? Teachers, in general, consider their syllabuses to be rational. Those responsible for the syllabuses draw on their common sense, on tradition and sometimes, though rarely, on psycho-pedagogical experience; these frequently have a bearing on the difficulties encountered by pupils taught by verbal methods in which notation is introduced before understanding has been ensured. Only a few pupils are entirely successful and the many who fall by the wayside have led to the belief that reform of the syllabus should be in the direction of reducing its content, delaying the

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introduction of certain notions, devoting more time to it and choosing the right moment for doing exercises.

The truth is far different, and it needs only a little goodwill to see that this is so. In conditions for the apprenticeship to mathematics such as have been described above, the vast majority of pupils from five to seven can study the set of rational numbers up to and beyond 100â&#x20AC;&#x201D;and it may be up to several thousands. In doing so they can master most of the operations with a speed and a delight which give clear evidence of the quality of their learning.

The syllabus is then clearly indicated. Every adult, including the mathematician, knows certain numbers and certain operations very well, knows certain interesting properties of other numbers, and with respect to the rest has no more than the ability to find some property when the occasion requires it. In just the same way our children must be allowed to investigate the numbers and operations which present themselves, gradually pushing further and further back the barriers which separate these three types of numbers. Moreover, as numerical experience has its interest in its applications, mastery in this field cannot hurt the pupils. But what is important in this training is that the attitude is mathematical from the start, and that the pupils study certain relationships, certain structures, which are first met in the sets of rods, then recognized as belonging to mental forms, and finally grasped as an activity of the mind.

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With the help of the Cuisenaire material we can win the race which our generation must run against time in order to prepare the world for a more intense and more extensive technological era. Moreover, the energy that is saved in the apprenticeship to arithmetic cannot fail to have its repercussions on the rest of school life, and therefore on society. There is already abundant proof of it in many schools where enthusiastic teachers have seen difficulties vanish, and now have at their disposal time which can be devoted to improving the growth of their pupils.

Moreover, the atmosphere created by the childrenâ&#x20AC;&#x2122;s joy in their mastery brings to education in general the contribution which the pioneers of New Education hoped to achieve through the lofty ideal of the teacher rather than through techniques.

The sound mathematical and psychological basis of the methods we suggest opens new horizons for the school work of the future.

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9 Introducing the Concept of the Set

This article first appeared in the U.S.A. in The Arithmetic Teacher, Vol. III, No. 3, April 1956.

Introduction If one is prepared to study the activity of the mathematician, not in its final form, but as it actually takes place, there are many new discoveries to be made. By adopting this attitude the problems met in the teaching of mathematics at all levels are seen in a completely new light. In this short paper I shall first consider the place of arithmetic within the activity of mathematics generally and then summarize a method I have developed with material inspired by the genius of a Belgian schoolmaster, Georges Cuisenaire.

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Characterization of the Mathematical Activity While it is true that every mathematician can recognize what is mathematical in the treatment of any subject matter, no definition of mathematics acceptable to all mathematicians could be proposed. My attempt to clarify this somewhat baffling position has led me to investigate what it is that characterizes the activity of the mathematician; this I have found to be an awareness of a particular type. We no longer need to contrast mathematical activity with other kinds of activity; we need only state what it is and every mathematician will recognize that this is, in fact, what he experiences when he is at work.

The first important point is to recognize that there are what we shall call virtual actions. Every human being is involved in a multitude of actions. When he stops to consider a position inwardly, for the purpose of selecting some aspect of an action as being preferable to others and deciding to embark upon it, he is substituting virtual actions for real ones. Moreover, virtual actions may lead to the extension of actual actions when these are replaced by internal processes. For instance, stringing beads is an action, while to imagine oneself doing this is, at first, to evoke the movement without actually carrying it out; to become aware of it as a possible action that can recur indefinitely is the virtual action which will serve as basis for the indefinite extension of addition of units.

Virtual actions have some of the characteristics of actions and hence of all biological and mental life. In them perception is implicit, and with it the dynamics of the mind. By becoming

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virtual, action extends its range, but the extension is not arbitrary. There were certain constraints inherent in the original action, and these now become potential. Thus, if for an actual staircase climbed we substitute the set of steps to be taken, the virtual actions we obtain will be arithmetical progressions, and nothing more. In fact, the structures contained in the virtual action limit the extension of the structures, and it is awareness of these structures that makes the mathematician.

All those, then, who are capable of replacing actual actions with actions that are virtual and of contemplating the structures therein contained, act, when they do these things, as mathematicians. All of us are, of course, capable of this but not everyone delights in that kind of exercise, and so all men, although potential mathematicians, do not become professionals or even amateurs in this field. Life offers many other attractions, both through actual and virtual actions, which involve awareness of things other than structures.

Structures and Relationships In the last fifty or so years mathematicians have come to see that they are essentially concerned with sets of elements in which they induce structures by means of relationships. What matters to them is to know the maximum field of validity of a statement and to discover the proofs which most adequately correspond to the propositions involved. They state, in fact, that the concept of â&#x20AC;&#x153;mathematical entityâ&#x20AC;? can be no more than a mental construct indicating a momentary halt in the process of discovering what lies behind a complex structure.

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But what mathematicians contemplate is mathematical situations containing a number of structures and relationships, themselves defined by certain relationships. There is an indefinite number of such situations. As an example, we can take the â&#x20AC;&#x153;geo-boardsâ&#x20AC;? which I have introduced as a simple but pregnant means to an understanding of what constitutes a geometrical situation. A board on which a lattice is marked can provide, when points of intersection are joined, various figures (polygons). These can be described and their number found. By concentrating on particular aspects of the figures we are able to select those which are congruent or similar, and those which are symmetrical or regular; we can calculate areas or look for concurrencies, and so on. The situation is mathematical through the fact that we are concerned with certain of the existing relationships, and not with the wood, the nails and the elastic bands which are the essential elements of the actual board. By our actions we create relationships in the situation, and we are doing mathematics since we are abstracting one or more relationships from the background of all the possible relationships it contains.

To differentiate between structures and relationships we could say that we are more easily aware of the latter and that when relationships become fixed in our minds we call them structures.

In relationships we are aware of the dynamic element; when it is overshadowed we see them as structures.

Circles are relationships, and we can conceive of relationships between circles, such as overlapping, intersecting, concentricity; from these we can extract new relationships, such as, for

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example, that the common chord of intersecting circles is perpendicular to the line joining the centers.

The vision the mathematical eye has of these things is a vision that can become available to others if they are made aware that they are aware of relationships. To my mind, the function of mathematics teaching is precisely this.

Arithmetic What awareness is peculiar to arithmetic? How are arithmetical situations created?

It is obvious that in arithmetic we are always concerned with numbers. The set on which we operate is a set of numbers, and the relationships that underlie the set are those which constitute what we can call “qualitative arithmetic”. First we have the awareness of sets and sub-sets, these being formed of the elements satisfying a certain relationship. Thus “equivalent relationships” (binary relationships satisfying the conditions of being reflexive, symmetrical and transitive), subdivide a set into sub-sets, or classes of equivalence, such that each element belongs to one class and one only. By selecting any one element from each class we obtain what is called a “quotient-set” of the given set with respect to that equivalence relationship. “Order relationships” (binary relationships which are transitive) structure some sets, and are more primitive than numbers. But the dynamic cause underlying arithmetic is the so-called “algebraic structures”. The set is such that “operations” exist in which to each pair of elements a further element corresponds.

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The way in which the correspondence is made forms a set of relationships fixing the algebraic structures. A very important structure of this kind is the “group”, which is recognized very early in childhood, and to which I shall return in a moment.

Arithmetical situations are created here through the Cuisenaire material.* This consists of a set of rods which is at once seen to be subdivisible into classes of equivalence (the same color or the same length), its quotient-set being formed of ten rods, one of each color. The whole set is ordered (the order relation being “greater than or equal to” or “smaller than or equal to”) as is also the quotient-set, but the latter is “strictly” ordered (“greater or smaller than”). The way in which the rods are cut allows an algebraic operation to be introduced corresponding to the action of placing rods end to end to produce new lengths; this can be noted as addition. In this way we obtain a set which, if we suppose the existence of an indefinite number of rods, is “isomorphic” to that of the positive rationals, that is, is in a oneto-one correspondence with it, and is such that the operations performed on a pair of corresponding elements in the set of rods, and on the set of rationals, give results which also correspond. This last statement requires proof.

Since the length of any rod or of any number of rods placed end to end is a multiple of the white one, we obtain the sub-set of rationals constituted by the integers when the rods are measured by the white ones. But if we choose to measure any length using as measure any other rod or length, we generate

* The detailed description of which is omitted here.

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pairs of integers which are true fractions. The operation of addition of the integers satisfies the same conditions as the operation of putting rods end to end and finding the ensuing total length. In the case of fractions this will become clear when we have taken one further step in the direction of arithmetical situations.

With the Cuisenaire material any given length can be constructed by a number of groupings of rods of various lengths.

These groupings we call decompositions, or partitions, of that length, and it can be seen that the length is not affected by any permutation of the rods in a decomposition (we call this the commutative property of addition). When rods are removed one after another, then, by knowing the original length and the length actually remaining, the particular rods removed can be discovered; in this way it can be seen that each addition corresponds to several subtractions. If the rods in a decomposition are of one color only we simultaneously obtain two factors, which are sometimes equal, for the number represented by the length, and hence several fractions of it and their value. If the length cannot be completed by repetition of one rod or group of rods, we obtain division sums. When the set of decompositions contains no row of one single color (or a row formed by a repetition of the same group of rods), that length is a prime number.

A set of decompositions is therefore a rich arithmetical situation. Clearly, the influence of the Cuisenaire material on arithmetic teaching is revolutionary. In the first place, it brings modern mathematics into the primary stage of schooling, and, in particular, the important recognition that fractions are 85

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ordered pairs of integers. Secondly, it substitutes for the study of numbers the study of the sets of their decompositions, making apparent the dynamic operations that structure these sets. As a result, addition, subtraction, multiplication, factors, fractions and division are all seen at the same time, and seen as being generated by the virtual actions of the mind on the situations.

Thirdly, through the presentation of isomorphic systems, results seen to be obvious in one appear to be also true in the other.

Color and length offer information that can be perceived, and the translation of this information into numerical language gives to this numerical language a dynamic which is far less apparent when the written notation alone is used. Fourthly, since the rods are not subdivided, they can represent a different value each time they are used as a measure for comparison with other rods.

It is this fact which gives the material its unique property of being able to introduce simultaneously both whole numbers and fractions, integers being seen as fractions whose unit of measurement is omitted and ignored.

All this is, of course, of tremendous significance in the learning and teaching of arithmetic. The several thousand classes in which the Cuisenaire material has been in use in the last few years testify to the fact that what earlier seemed impossible can now be easily achieved. Children of six and seven are thoroughly familiar with their tables; children of five conceive of, and compare, fractions easily and accurately; children of eight solve simultaneous equations, and at ten they understand permutations and combinations which they form and analyze themselves.

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A detailed account of my work with this material in several countries has been recorded in other publications.* The research stimulated by the introduction into schools of Cuisenaireâ&#x20AC;&#x2122;s Numbers in Color can be measured by the necessity felt for the creation of new international journals for the publication of material resulting from its use.

Conclusions It has not been possible in this short note to elaborate any of the points I have made. This could be done only in the form of a book. It may be felt that what is ultimately of prime importance is the raising of the level of teaching in schools, and that modern mathematics cannot be introduced into the syllabus of the early years while teachers of the first grades are insecure in their own knowledge of what they have to teach. This situation is universal and I have found that I can give to teachers the training that will enable them to recast the content of their syllabus and their methods of teaching, through intensive refresher courses lasting from one to two weeks during which I use with them the material they will use on their return to school. The ready response I have had from teachers in primary schools in several countries gives grounds for optimism and supports our hope that as a result of a sustained effort we shall, in a few years, see arithmetic enjoying a popularity equal to what it has at present, with number fans.

* Much of this is reproduced in this volume.

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We now know where the trouble lies and where the remedy is to be found. Pupils of all ages and ranges of ability have responded to my teaching with joy and a real understanding of the issues, which takes them to a much higher level and is much more abstract than is usually the case. Other teachers who, like me, have had faith in their pupils and dared to stretch their minds, have had a similar experience. We are no longer in the trial stage and can look to the future with confidence.

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10 Notes on a Radical Transformation in the Teaching of Mathematics

This article first appeared in the U.S.A. in Main Currents in Modern Thought, New York, January 1958.

In a world that has undergone such great technological change in so short a time we are confronted with pressing questions in the field of education. Can we afford to continue to use ways of teaching which arose in mediaeval Europe and were essentially derived from the Greeks and the Jesuits? Why has teaching remained so close to the ancient pattern when education, in the hands of great pioneers, has shown infinite possibilities of growth? Improvements in classroom design and in school buildings, the lavish use of color in school text-books, make only a surface contribution to the real challenge, which is that of creating a generation of young men and women who are at the same time competent, socially integrated and free in spirit.

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The methods we have inherited in the West succeed in making the intellect sharper and more penetrating, but they fail to provide a succession of spring-boards by means of which the mind can become ever freer of the restrictions it meets. The old methods are directed towards disciplining the mind; what is needed today is an education that leads to greater awareness of what constitutes the mind and of how to master the dynamics of thought, of the emotions and of creativeness. We propose here to discuss this radical transformation in the aim of education in relation to the strictest of all subjects, mathematics, and to do so in practical terms. This discussion is most relevant to the situation in which education finds itself today. We may note that in the United States, the only country where it was permissible for children not to take mathematics as a school subject, the clamor for more mathematics is now loudest. But in other countries too the small percentage of those who benefited from their mathematical studies posed a very serious problem to educators and administrators. It thus seems that the pedagogical question confronting all the Western nations today has two sides to it: one concerned with teaching methods that would perhaps permit more and more children to acquire more and more mathematics, and the other concerned with what sort of mathematics to teach in our present world, so full of complicated challenges.

It is my intention to examine the techniques of the process by which greater mastery and freedom in mathematics can be achieved by all children to whom the opportunity is given. But first I should emphasize that mathematics has a unique role to play in education, because its main function is to prepare everyone who cares about it to acquire its techniques for the task 90

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of making mental models of situations, including in them the most relevant data, and deducing from these models conclusions that were hidden but which are needed in order to forge ahead. In short, mathematics is much more than a tool for calculating oneâ&#x20AC;&#x2122;s pay envelope, oneâ&#x20AC;&#x2122;s tax situation, or oneâ&#x20AC;&#x2122;s profit margin. It is a power given to the individual so that he can find more in what he meets. This power is the birthright of every one of our children, and schools should make it accessible to all.

Before entering into details, we must familiarize the reader with certain concepts that are essential in our thinking, one of which is the concept of dialogue between the mind and reality (the latter including the subject himself as a consciousness). It is by confronting the whole of the mind, mobilizing the energies of spirit, with situations met, that growth in awareness and knowledge takes place. It is obvious that when alert minds meet the same situation (roughly defined) they find in it things that are different. This could not be so without the presence of a third term which is made obvious by what we have called the dialogue: the mind and the situation engage in an intercourse in which both play a part. When the situation is not external, but involves the content of the mind itself, there is still dialogue within the mind itself, and learning is then the deepening of awareness. The situation is shared, but since minds vary, dialogues differ.

Since we consider situations as data, we need only say that a configuration, or a set of internal relationships between things, people, events, forces, and so on, is assumed when we speak of a situation. The configuration is natural to the situation and need

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not be specified, or even known, yet we all assent to it as implied and inevitable. Hence, it is a basic concept.

Without situations, taken in this sense, reality would be meaningless to the mind, and we shall assume that the meaning of both “situations” and “reality” is as intimately known to every reader as is the meaning of “mind”.

Growing up is the result of the use of time in successive dialogues between the mind and reality, and greater awareness of one’s relationship to reality is concomitant with mastery of skills. These skills are moreover the outcome of the dialogue to take place, and maintain in the proper channels the energy necessary to make them available. By looking we learn to look, by hearing to hear, by writing to write.

Thus we have a reality that is dynamic because it contains minds continuously engaged in changing themselves and in forming relationships between themselves and the rest of reality. This reality can be envisaged as being composed of situations, and the learning process can be schematized as a dialogue of minds and situations, each changing the other.

Although in all its dialogues the mind is the same mind, some situations it meets may be mathematical. A new definition, in psychological terms, of the word “mathematical” is of essential significance here. Situations become mathematical, by definition, if the mind perceives in them only relationships, and ignores all the other possible attributes. But since relationships are met in situations, the mind is also aware of the various

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dynamics connecting these relationships. Mathematics is therefore the realm of experience in which what is contemplated, spoken of or written about, concerns relationships and their dynamics, entities being produced from time to time when the mind ceases to relate them to each other in order to move ahead. Objects are not the concern of the mathematician. When the word is used it refers to a class of unspecified things. x = 3 is not a mathematical statement, although it is expressed in a notation created by the mathematician in order to meet his challenges. A mathematical statement is of the following type: â&#x20AC;&#x153;If we consider a class of relationships submitted to the effect of a relationship we find in the new situation the following relationshipsâ&#x20AC;?.

Having clarified my idea of what constitutes the background of learning and of mathematics, I now consider how this view can be translated into a program of study for all pupils.

In my opinion, mathematics has never been taught in school.

What has been taught is bits and pieces of knowledge, and it is a marvel that, in spite of this fact, some people have become mathematicians. It is not to be wondered at that the majority of our clever pupils understand nothing of the subject and quickly drop it. If they are to be won over to mathematics, all that is needed is that we expose them to true mathematics. Since mathematical activity is natural, there is no more reason to fail in it than in walking or talking, when the necessary mechanisms are present.

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My approach to the teaching of mathematics then will be as follows: I shall teach algebra before arithmetic (arithmetic no longer being a collection of recipes for the solution of impractical problems, but the study of the properties of numbers), giving the pupils the full fruits of the thinking of mathematicians of the last eighty years, and exposing them to structured fields of study. Thus the emphasis on Euclidean geometry will disappear, and the false sense of rigor which was considered to justify that study will be replaced by a progressive widening of the fields, a deeper understanding of the assumptions and clearer statements of the results obtained.

With all this expressed in an adequate notation, what the pupils produce looks strikingly similar to what mathematicians publish. We shall have satisfied criteria which are at one and the same time psychological, pedagogical and mathematicalâ&#x20AC;&#x201D;and this is something new.

We must now justify our statement that algebra will be taught before arithmetic, and show how it can be done. Algebra is defined as the set of propositions that follow from the awareness of the effects of one or more laws of composition on the elements of a set. An algebraic relationship is one in which, under certain conditions, two elements of a set are substituted by another element of the set. Thus addition and multiplication are algebraic relationships in the set of whole numbers.*

* Note: not all relationships are algebraic: for example, comparison of the magnitude of two elements, or the formation of ordered pairs, are not of that category.

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It is obvious that by giving children a set of objects upon which it is possible to define an operation, we can enable them to discover what propositions follow from that simple fact.

Fortunately, a Belgian teacher, Georges Cuisenaire, has produced a set of colored rods where the operation of placing them end to end is “isomorphic” to addition and a multitude of propositions can be obtained from such simple situations as having two equal lengths and forming any length by placing rods end to end, without any mention of numbers. This can easily be done by children of five, and they have no doubts about the propositions obtained. For the mathematician this is algebra; for the child it is a meaningful game which can be varied indefinitely.† Before describing the method used we must first explain that the Cuisenaire rods are 1 square centimeter in section, from 1 to 10 centimeters long and are colored according to a scheme developed after twenty-two years of experimentation. This material represents a unique contribution to elementary mathematics learning.

Since color is an indication of a particular length, the set of rods contains sub-sets which are distinguishable by either color or length. The presence of a double attribute in the set offers two terms of reference: a red and a yellow rod end to end are equivalent to a black rod, or to a green and a pink end to end.

This equivalence of length is also talked about using color. But it can at once be seen that, in terms of the operation of placing

† See Mathematics with Numbers in Color, Book I, Part II.

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rods end to end, red and yellow is equivalent to yellow and red, and we thus discover a property (commutativity) of the operation, and that any length can be formed of two or more rods and conversely (this yields associativity). When several rods of the same color can be used to form a length, we can introduce a new arrangement of rods, side by side, and find that another operation (which can be called multiplication) has made its appearance. The relationship between these two operations is provided by distribution of multiplication with respect to addition. Its inverse is called factorization.

All this and much more becomes obvious through the use of the rods. The knowledge of how things are done belongs essentially to the realm of the algebraist, and the early discovery by the pupils that the action of putting rods end to end does not depend on the particular rods, that the commutative law is inscribed in the patterns whatever their length, that associativity results from certain equivalences and not from the elements involved, plunges them right from the start into this realm in which the tools because of their range are very powerful and where the mind is unrestricted by particular connections which must be memorized as such.

When the algebra of the set, that is, the operations introduced and their inverses (when these can be defined), is thoroughly mastered (which takes a few weeks with children of five or six), measurement of the rods one with another will give rise to a new language for the propositions already encountered, and new properties will result from the new structures involved. It is the language of numbers, and we can at this stage embark on the

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study of the set of rational numbers and their distinctive and important properties.

Here again, since we are concerned with psychological and pedagogical criteria at the same time as with mathematical rigor, we approach the study of numbers without prejudices: our pupils study the new properties which result from the new structures and use the algebra they have acquired in order to pass from one cardinal number to those that are linked with it within a certain range. The addition of 1 is not the only key to the formation of the set of integers. This set becomes the union of all sub-sets that can be studied by all the means at our disposal. In particular, by using an operation such as doubling we produce sub-sets which are spread over the integers in a very different way from the uniform but uninteresting method of addition of 1.

In using our freedom with respect to the way we obtain the integers, we meet many more of their properties and the set of integers becomes more and more fascinating. Congruencies* emerge naturally at a very early stage and commutative ringsâ&#x20AC; appear almost from the first, since so many examples are met.

* Two integers are congruent with respect to a third integer if they have the same remainder on division by the third; in particular they are congruent if they are both divisible by the third.

â&#x20AC; A commutative ring is a set in which are defined two commutative algebraic operations (such as addition and multiplication).

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Composite and prime numbers are studied as a matter of course, appearing as naturally as counting disappears from the study of arithmetic.

A further important consequence is that the solution of problems is no longer sought by rule of thumb but by scrutiny of the structures involved, and the algebraic solution comes before the arithmetical one, as common sense would suggest should be the case.

When algebra precedes arithmetic, freedom comes into its own and the effective strait-jacket of discipline is forgotten.

Elementary mathematics learning then serves the true education of the child. We use actual actions upon actual objects and ask the appropriate questions, thus giving to every child every chance to see, rather than leaving him to guess, what we mean by them.

But mathematics is not reducible to operations and numbers: its spatial aspects must also be considered.

Here once again we must recognize that spatial relations are not all metric, and that we must first discover the structures which form the background of the mathematicianâ&#x20AC;&#x2122;s spatial judgments.

Invariance and preservation of properties by certain transformations* have been studied by geometers, especially

* As an example, in the distortion of a torus into a sphere with a handle, the fundamental characteristics of the torus (that it is a closed surface surrounding a single hole) are preserved, though metric properties are drastically altered. Analysis Situs is the study of such invariant properties and of the transformations which leave them invariant.

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during the nineteenth century, and in 1874 Klein defined a geometry as a set of properties invariant with respect to a group of transformations. The simplest group is that of displacements, and Euclidean geometry is the system of properties of two- or three-dimensional sets invariant with respect to similitudes.

Topological properties being involved in all spatial relations, it is now a platitude to suggest making our pupils aware of their presence as well as of other properties. As yet there has been no publication of any systematic presentation of spatial relations at the elementary level parallel to the one described above with respect to algebra and arithmetic. I am at present engaged on such a task and hope soon to be able to publish the result of this work.

What has so far been developed is a study of finite geometry making use of my geo-boards (which are finite lattices made material by using nails on boards and on which colored rubber bands can be stretched). Since the boards can revolve and be moved about, the group of displacements affects all figures and properties. The main virtue of the geo-boards is that they provide a great variety of geometrical situations, which can be investigated in the order in which they are met, and not in the order of Euclid or his followers which makes it preconceived.

Geometrical situations can be so easily produced that the 25-nail rectangular lattice alone presents over a hundred of them.

Although the geo-boards only allow an investigation of metric spaces (that is, not more than was required by schools) the few lattices presented have a valuable contribution to make since they permit geometrical education as opposed to the geometrical conditioning now taking place.

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In the attempt now being made by the International Commission for the Study and Improvement of the Teaching of Mathematics to rewrite the whole of elementary mathematics for adolescents in the perspective of modern thought, we may find a first answer to the problem of substituting for the geometry which is at present supreme in mathematical education, truer mathematical thinking which consciously uses what is important in it and discards what is sterile. The work done in algebra and arithmetic for the first grades is most encouraging, and it is hoped that the secondary stages will be as effectively dealt with. The problem of teaching mathematics to all, in our highly technological age, will then be solved.

It should be said in conclusion that those of us who are engaged in the radical transformation of mathematics teaching are making a wholesale attack on the problem and are seeking a solution applicable at all levels of learning. Aware that the wrong track has been followed for centuries only through ignorance of fundamental facts, and confident because of the successes already scored, we look forward to the time, not too remote, when the only mathematics teaching problems we are going to meet will be new problems, the old ones either having proved to be only pseudo-problems or having been solved.

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11 Thinking Afresh About Arithmetic

This article first appeared in the U.S.A. in the Arithmetic Teacher, February 1959.

When so many people have already been engaged in examining the basis of our knowledge of number, there would seem to be little chance of finding something new to say on such a venerable subject. Still it is my contention that we have barely started examining the field and that at least one generation of research workers will be needed to clear the ground that I shall survey in this article.

The critical study of the conceptions of number put into circulation in the last seventy years by Frege, B. Russell, A. N. Whitehead, Hilbert and others, and more recently by Piaget, may be put aside for another occasion or left to other writers altogether. It may be that the main difficulty has been the attempt to define number, thus assuming that such a concept existed and could embrace all numbers. Are we not here facing a

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difficulty comparable to the definition of man? Perhaps it would be easier to define each number using some recurrent pattern of thought, some process, akin to the complexity encountered. For teachers of elementary grades that means that we have to ask ourselves whether counting is a good basis for the presentation of number and its formation. Counting sometimes means reference to a temporal sequence of sounds often repeated and memorized; sometimes it means (in counting objects) that we recognize that we pass from one sound to the next, adding one (assumed to be one and the same unit); sometimes it means that we can label in a particular way one attribute of a collection which then appears as of the same level of abstraction as other attributes, such as shape, color, consistency, and so on.

So long as we hold that we must use counting in the formation of numbers we cannot conceive of any other means of introducing arithmetic to our very young pupils. It has been long recognized by individual parents and psychiatrists that anxiety appears in children when they fail to comprehend arithmetic, and that mathematics is the school subject creating the greatest amount of passionate opposition, often of a lifelong duration.

Are we not to learn our lesson about the nature of mathematics, children and learning, from this deep emotional involvement of people with mathematics? I maintain that learning mathematics is a very natural activity, comparable to fundamental biological activities such as walking, talking, driving, and so on. It is concerned not with knowledge related to memory, but with knowing how, and with biological organization linked with reflexes. By making mathematics dependent on memory, as we do through counting, tables and rules, we denature mathematics

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and force the child to meet reality clad in a garment that neither belongs to reality nor fits it.

Do we adequately appreciate the aptitude children must have in order to learn (so early) to make the sounds people around them make and so soon to be able to use them freely to express what they want? In learning so quickly to talk so well and so freely, children show us what they are capable of doing with concepts.

They certainly learn at an early stage to suspend their judgment before deciding which object or image belongs to a class they are considering, thus demonstrating that they are thinking in classes even while perceiving individual objects. They can be said to meet the general first, then by growing in experience, they fuse the general structures into one entity to form particular classes or objects.

This is a most important observation and should become better known for, when better understood, it will change our outlook on children and education. Once made, it leads us to a multitude of improvements of techniques of teaching and to a saving of time, and time is precious as it is all we have in life. We should not throw it away as we so often do at present, particularly with our pupilsâ&#x20AC;&#x2122; time. Speeding up the learning processes is not a fancy of one educator. If it can be done by reorganizing our resources and not by cramming, it means we have better understood part of nature, of reality, and can wholesomely serve the purpose of growth, which is the aim of education. If we have not thought in that way so far we should begin to do so now.

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Another important observation is that, for each one of us, experience is one indivisible whole and takes place in the total system which is our self, our body-mind-spirit, and we can use this observation to integrate all our powers of knowing. To say that mathematics is concerned with intellectual rigor is to forget that we feel mathematics as well as think it; that it often is the awareness of something within us that is at the origin of our awareness of a concept which we utter as words or explain, using our images. All mathematical discoveries of importance can be traced to a dynamic alteration within one mind of existing organized images, or ideas.

Thus it is wrong to separate experiences into compartments, to say mathematics is abstract, as if music or talking were less so, or to say mathematics is rational (meaning by that that it must be formalized before it acquires its full status), as if reason had in mathematics, a favorite garment and did not work equally well in law, cooking or any other human endeavor?

Since emotions can block, they can also motivate and assist.

Since children have to mobilize the whole self over and over again, then in order to organize the new and transient in that which they meet all the time, we must make use of their affectivity in our presentation of mathematical experience. We must use all the components of self (whether known to us or only intuitively suspected) to help them enter as wide a field as possible, putting to them an ever changing field of experience that challenges them incessantly. If mathematics has not disappeared from the face of the earth it is because it is forever being renewed, as are all human arts, by the constant interference of the mind through a new point of view. That is 104

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proof that we can make use of ever-changing situations to challenge our pupils. But change of viewpoint does not mean by necessity change of material. Even today the set of real numbers remains the most investigated field of study. The approach in mathematics is not, in fact, the discovery of what is hidden, but the introduction of new light that reveals the unsuspected. The lighting is created by minds and it carries with it the possibility of a different structuration of reality. So mathematics is a constantly renewed dialogue that returns to the same entities to show that they also contain this or that.

We say 1, 2, 3, . . . . n, . . . is the set of integers. It seems such a simple set, but only because we look at its first few members; it is in n that the mysteries are hidden. Who can conceive of a number of 10,000 digits? The day we develop the ability to see large numbers, not a discovery but a revolution in number theory will take place. Our studies intimately reflect our standpoints, and we cannot yet say whether the edifice we call mathematics is of this or that architecture. Our children, to be said to be educated, must learn to become aware that their horizon varies and widens as they grow up and, commensurate with the increase in their powers, the field they can encompass and upon which they can act increases.

Our traditional arithmetic is, in all its aspects, paralyzing and uninspired and therefore pedagogically wrong. To renew it we must attack ourselves and our conceptions and recognize that Frege and Russell may be wrong concerning what the foundation of mathematics is.

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In my own life came the shock of finding out that Georges Cuisenaire, who knew no mathematics and who had never read anything about its foundations, was the one to show us the way of making children transcend the traditionally revered limitations. He used the model of musical experience and from it produced the keyboard on which every child can play not music but variations that are mathematics. That he came from an unexpected direction conforms with what we know of genius and the light it throws on things. What no mathematician, logician, philosopher could do, a school-master did; for this I, and teachers generally, should forever be grateful, and we should all learn from this that since life is never ending, always renewing itself, our education should begin to resemble it through our work and endeavors.

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12 Reality and the Learning of Mathematics

This article first appeared in New Zealand in Education, Vol. 10, No. 5, June 1961.

Reality is not only social living with its demands for shopping, travel, payment of taxes, and the like. Reality includes the spiritual universe, intellectual experience, and the unknown held hidden in the future descending upon us.

So far the teaching of various subjects in the school curriculum has been restricted to doing to others what was done to us, and teaching mathematics has been the same steep uphill climb exhausting so many that very few people today have an appreciation of what could be meant by mathematical reality.

I contend that a thorough revision of our attitudes towards both mathematics and its teaching could, for the vast majority of the worldâ&#x20AC;&#x2122;s population, result in reasonably smooth sailing in the beautiful ocean of mathematical experience. Indeed, the major 107

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obstacle today preventing this from being achieved is the lack of co-operation of teachers and educational administrators with reality. That is to say, in no time things could be changed radically if we transformed ourselves into people of goodwill meeting reality and working in the light of it.

In this short article I can only put forward a few of the many arguments available to convince the reader as to how it could be done. I only hope that on understanding these he will move on to seek for other reasons, and work on them, in order to acquire a still better grasp of the vaster meaning of the reality of mathematics.

There are at least four kinds of evidence to be considered here: (a) introspective evidence that may show us how our fears, ignorance, obedience of rules and regulations, complacency, and beliefs act upon our attitudes and determine how we meet what lies in front of us; (b) the objective evidence of our pupilsâ&#x20AC;&#x2122; mistakes which are repeated year after year in classrooms everywhere, irrespective of the background of the pupils, the personality of their teachers, and other factors such as climate, language, and so on; (c) the role of tradition that affects all that is done in our schools and prevents us from looking at things directly; (d) the experimental evidence that brings to our notice facts quite different from those we have heard of and read about.

Though there are still other kinds of evidence to challenge our complacency I shall restrict myself to these, and even then shall be leaving (a) to the reader, and suggest that for (b) any teacher may gather together a collection of mistakes, analyze them, and

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be familiar with their causes. He would discover, as I did, that many of the mistakes allowed to pass as being due to the dullness of children are in fact due to bad or wrong teaching.* For example, infants’ teachers know that mistakes are more frequently made in additions of the form 3 + 9 or 2 + 7 than of the form 9 + 3 or 7 + 2. Why? The answer that comes immediately should be scrutinized rather than accepted at once.

As for (c) and (d) I may say something here that will suggest where improvement may come from, and which will not appear as substituting opinion for opinion. It is traditionally accepted that place value is important in the study of arithmetic, and time is set aside for this in schools. The reading and writing of the names of numbers is considered to be a way of learning them.

But one does not know a flower or a person just by knowing their names. The construction of a name for a number depends on notation and other deeper notions such as the use of zeros for the empty places, the scale of notation accepted (decimal or binary, for example), and the algebra of entities called polynomials (for example, 2735 is equal to 2000 + 700 + 30 + 5 or 2×103 + 7×102 + 3×10 + 5, called a polynomial in 10 of degree 3) which make operations on numbers take the form we know (the addition of units up to ten, then of tens up to ten tens, then of hundreds up to ten hundreds, and so on). If we really wished to teach the written notation with all the deeper notions involved in it, a very wide experience would be required before it was properly understood. Thus many of the components of the situation are left out and pupils are given practice with addition

* See Part I Chapter 3 of this work.

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in columns till they can do the sums automatically. We form number names by using groupings if we are using counters, or bundles if we use sticks, or lengths with Cuisenaire or Stern materials.

Now let us depart from tradition at this point and tell ourselves that we have a challenge in the reading and writing of numbers that is to be met as such, and for the time being let us not be concerned with an understanding of the deeper notions, which are beyond us in any case as soon as we consider numbers of a certain magnitude. If this is agreed, then the problem reduces simply to this: if we can read and write all numbers of up to three figures, then we can also read and write a number composed of any (reasonable) number of figures by simply introducing, say, the names of a few commas. Thus, if we can read 5; 25; 325; and if the comma that appears now is read thousand, we can easily read 4,325; 14,325; 614,325 and the like. If the comma next introduced is read million, we can read 1,614,325; 71,614,325; 871,614,325 and the like. If the third comma is read thousand million (billion in the United States), we can read numbers whose names sound like 123,987,654,321 and so on. In this way the problem of the reading and writing numbers has been made so simple that six-year-olds delight in writing such large numbers from dictation. If the readers of this article try it out with their pupils they will find that it works.* Of course, children do not know (any more than I do) what these very very large numbers are which they certainly can read and

* See Mathematics with Numbers in Color, Book II, Part VI

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write. But they know their names as I know the names of stars about which all I can say is that they exist in the sky.

Putting the reading and writing of large numbers at the disposal of small children is a breach with tradition. Here is an important consequence that follows from it. The fact that we have met the reality of the situation has at once, like Sesame, opened up a treasure of experience carried within the notation itself. As it is structured according to the deep notions mentioned earlier, we have given the learner the means of reaching swiftly a stage which would otherwise have taken a long time to reach. He can now, as soon as he masters the essence of an operation, see that it can be extended to numbers beyond those on which he understood that operation. He does not have to wait months or years to find that out. It is there almost at once.

In the study of subtraction let us help our pupils find (and this is yet another breach with tradition) that so long as one number is larger than another their difference can be found, and that it can be found by one and the same method whatever the length of the number names involved. This treatment, though new to the reader, comes from a very simple observation making any of the famous (or infamous) methods of subtraction advocated in education altogether unnecessary.

Indeed the observation is subtle yet very simple. The statement 2 + 3 = 5 is complete and ends there; 5 = 2 + 3, however, is incomplete, and asks for other statements such as 5 = 4 + 1 = 3 + 2 = 2 + 1 + 2 = . . . . Similarly, 5 â&#x20AC;&#x201C; 2 = 3 is a complete statement whereas 3 = 5 â&#x20AC;&#x201C; 2 is not. Here we have an infinite sequence of

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equivalences, such as 3 = 5 – 2 = 7 – 4 = 17 – 14 = . . . . Thus every difference is equivalent to others and the simplest of all is of the type 3 = 3 – 0, where a zero appears to mean that there is no subtraction to be performed but rather only the reading of the other number, which is the answer.

If the learners have knowledge of the number names and have had practice in finding subtractions equivalent to other subtractions, then, as we can see, the learning of subtraction no longer requires that the structure of numbers be studied. Nor is it any longer necessary to separate “hard and “easy” subtractions. All are treated as having the same status, that is, easy. Thus 1001 – 836 (which traditionally is a “hard” subtraction) is equivalent to 201 – 36 by dropping 800 from each number, and to 205 – 40 by adding 4 to each, and to 265 – 100 by adding 60 more to each, and to 165 – 0 by dropping 100 from both. Another pupil may add 64 to each at the beginning, getting 1065 – 900, and then by dropping 900 from both, get 165 – 0 at once. In my various publications readers will find more on this topic to make it much more acceptable. What I have done here is to give two examples to show how, by leaving tradition aside, learners and teachers might be helped in their tasks at school. As for multiplication, long division, fractions, and decimals, there are reasons inherent in mathematics for my treating them in a new way. I have shown this in several texts written for children and teachers and will not dwell upon them here.*

* Mathematics with Numbers in Color, Books II, IV and V.

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But I must add something about (d) above. Experimental work carried out over a period of years with a considerable number of pupils in many countries has convinced me that we cannot claim that, in our schools today, we take children fully into account.

They are, however, a part of the reality of the situation. I should like to mention that as soon as I take them into account they force me into making such changes in my procedure that almost everything I had heard about them is contested, and generally found to be unjustified. For instance, we have all read in books that with slow learners we need to go slowly. I found, on the contrary, that the slow learners needed over-stimulation. In fact they responded much better and seemed to retain things incomparably longer when I went faster with them than is usually believed that they can go.

Again, we all learn that to teach well means to meet one difficulty at a time. So we subdivide the subject matter in atoms, and give it to the learners one atom at a time. This, I found, leads to a starvation of the intellectual needs of most of our children, who are much more alert and powerful when faced with large dynamic units that support each other and thus reduce the efforts of memorization while associations and coordinations are increased. By giving to the learners, say, the following set of experiences of the number 30, we have no fear that retention will be a problem: 30 = 2 × 3 × 5 = 5 × 3 × 2 = 2 × 5 × 3 = 5 × 2 × 3 = 3 × 5 × 2 = 3 × 2 × 5 = 2 × 15 = 15 × 2 = 10 × 3 = 3 × 10 = 5 × 6 = 6 × 5

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In so doing we meet very many aspects of 30 at the same time. It is easily understood that the web of relationships found round a number is stronger, better knit and more readily evoked than the number itself in isolation. The evidence all suggests this to be the case. It would be to the benefit of all were the readers to recognize this after experimenting themselves and finding out for themselves the value of such a procedure.

The Cuisenaire materials are a very good model for indicating these connections between numbers. Also, symbolic thinking in children is vastly increased when using them. While trying to understand the properties it becomes clearer that any one of the rods can have many names from the fact that it can participate in many relationships with other rods. For example, the tan rod is one by itself, two in relation to the pink, four to the red, eight to the white, one and three-fifths to the yellow, and so on. On the level of symbolic thinking carried still further, a rod may also represent an amount of money, a percentage, a beam in a building, a member of a proportion and so on.

Children show us that we have clumsily interfered with the expansion of their abilities by starving them (in good faith and 114

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believing it was done for their own good), and molding them rather than learn what they can actually do with each of the challenges in front of them. By letting them teach me some of the things they can do I came to some shattering conclusions which I shall summarize here, leaving to the reader to see whether I have been fooled by my beliefs or whether I have been privileged to meet the reality of the situation.

1 Children between five and seven can cover with full mastery and understanding the work in mathematics supposed to demand the time between the ages of five and eleven. This includes the four operations with whole numbers, mental arithmetic with numbers up to the thousands, fractions as operators, problems involving money, lengths, areas and volume both in the British system and in the metric system, problems involving sharing and proportions, and other things as found in my Mathematics with Numbers in Color Books I to III inclusive (except for one or two sections).

2 Children between seven and twelve can easily assimilate subject-matter now taught in secondary schools, or not taught but nevertheless advocated by â&#x20AC;&#x153;progressiveâ&#x20AC;? groups in the United States, Europe and elsewhere, such as equations, different bases of numeration, geometry, and so on. I include the algebra of sets, permutations and combinations, and progressions for children of this age.

3 Only a very small percentage of the pupils in any one age group experience difficulties compared with what our traditional approach has led us to expect. Previous investigators engaged

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themselves in measuring the capacity of children with types of tools that created many failures. They could not fail to come to the conclusion that only a few children are endowed with mathematical ability. Now that other means exist permitting teaching to be subordinated to learning, it is being proved that the truth of the matter is quite different. All those who are ready to take children into account are discovering that, on the whole, the vast majority of pupils have mathematical ability.

Just as the preconceived idea that the earth is flat must be dropped before it can be thought of as spherical, so we need to abandon our preconceived (formed or borrowed) ideas about children before we can see them as they are. Our children in our democratic society now need to be vindicated as persons, just as women were not so long ago. Whenever the opportunity is given children present to us their true nature, a sight which is moving and delightful.

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13 A Matter of Relationships

This article first appeared in The B.C. Teacher, British Columbia, Septemberâ&#x20AC;&#x201C;October 1957.

All teachers will agree with me that their teaching of arithmetic is based essentially on the use of counting, which would appear to be the simplest of all ideas. Is there not a logical foundation for saying that 6 + 4 = 10 is true because 6 + 1 = 7, 7 + 1 = 8, 8 + 1 = 9 and 9 + 1 = 10, hence the answer? Yes, but are we to be content with that? We do not stop at proving that 6 + 4 = 10, we also want it to be remembered, and that is where logic no longer operates.

6 + 4 = 10 is only found to be true if I resort to counting; if you ask me what it is, I shall use my fingers, or counters, or whatever I have. But that is not what you want; you want me to say at once that six and four makes ten. Counting is not to be used because it wastes time. Why, then, did you â&#x20AC;&#x153;teachâ&#x20AC;? me to count if I am not allowed to use it?

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“I taught you to count”, you will say, “so that you would know your numbers and understand addition. Now I want you to drop counting and ‘know’ addition bonds.”

“If I said I know addition bonds, would you believe that it is because you taught me to count?”

“Yes, how else could you know that 6 and 4 makes 10?”

“If I said I just know it because I can remember strings of words, then would you be satisfied?”

“No, because I want you to understand what you do.”

“Then I must use counting to find 6 and 4 makes 10.”

“No, since you should, by now, know it.”

“But if I don’t, may I count?”

“You may, but you should know that 6 + 4 = 10 soon, otherwise there must be something wrong with you.”

“Similarly, you ‘teach’ me the multiplication tables, and I can recite them. But you want me to give the answer at once that 7 × 8 = 56; if I say 1 × 8 = 8, 2 × 8 = 16 . . . 6 × 8 = 48, 7 × 8 = 56, you get cross with me and say I should “know” it. Is it not knowing if I am able to stop at 56 in the sequence of the tables I

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so laboriously learnt? You wanted 7 × 8 and now I am telling you that it is 56.”

“Yes, but you should know it by now, otherwise there is something the matter with you.”

“Then why did you ‘teach’ me the tables if what you want is the products?”

“So that you would understand that we obtain the answers by always adding 8 to itself.”

“But that is precisely what I did to find 7 × 8 = 56.”

“Yes, but you must not use the tables any more, you should know the answer at once.”

“But why, then, did you ‘teach’ me the tables in the first place?”

It is obvious that we can multiply the examples of similar teaching situations which show that as a profession we are not very clear about what we are doing nor where our pupils get lost when confronted with our contradictory requirements which often make them give up altogether.

This is not the place to examine our syllabus critically and pinpoint the confusions we create, but it would pay all teachers to do it. All I can do in such a short article is first to tell teachers that they can in fact avoid confusing their children, and secondly

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to explain how, if we want to teach arithmetical bonds, we must not start with logic but with action, action that is self-convincing and of the kind which is the backbone of our belief in logic* Thanks to a stroke of genius, a Belgian, Georges Cuisenaire, has provided us with a means that makes arithmetic teaching a reasonable activity. I can confidently say that if there are problems now in arithmetic teaching they are new ones, since all the old ones have been solved, and all children can reach a very much higher level of achievement than we ever dared dream of before. But what is Cuisenaire’s idea, and how does it work?

Cuisenaire tried to bridge the gap between the so-called concrete and abstract levels of arithmetic and in the way many teachers do he used blocks, which he colored according to specific ideas of his own and which he did not mark into units. This very

* In fact teachers would do well to meditate upon their teaching and see when they mix the various parts of the learning process and put in front of their pupils an irrelevant remark. Here is one example. A child may know quite well that three and three makes six, yet in writing it, he may produce the following pattern: +3=∂ (as teachers often notice in infants’ work). The teacher who forgets that the child is involved in a recording of arithmetic when this is written may want to “correct” the wrongly written 3 and 6. This zealous remark may be an interference at first which, I feel, should be left to a later moment when the child is reviewing his work and can then be made aware of writing, rather than of arithmetic. Young children are learning to read, to write, to add or subtract, and we are asking a great deal of them all at once. In arithmetic, what matters most is the mental process, not the marks on paper which, at first, are there only to please us. When teachers realize this, rapid progress can at once be noted in the pupils, for arithmetic is easier than the complicated notation recording its findings. In any case, it is better to know that in learning mathematics there are several activities, stressing each at its proper time.

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simple thought did the trick: because of the colors, children were taught from the start to work with relationships, with bonds rather than with specific numbers, and these bonds became numerical when special comparisons were made. Thus the red family of blocks, or rods, (red, pink and tan) is such that the length of the first is half that of the second which is half that of the third; the blue family (green, dark green and blue) is such that the length of the first is half that of the second and one third of that of the third, the yellow family (yellow and orange) is such that the yellow is half as long as the orange. The children are given a number of each of these rods and also some white and some black ones. Each white rod is half each red and a third of each green, while the black rods are seven times as long as the white.

Color Relationships and Values It is clear from this description that many relationships are contained in the way the rods are colored, and that each rod has different values according to which other one it is compared with.

In fact each rod is a complete entity and its color distinguishes its length. What children do, after playing with the rods as building blocks, is to abstract in turn the relationships contained in the sub-sets of the rods. Because the choices of sub-sets are many and the relationships they contain also large, it happens that with the Cuisenaire rods all the ideas children have to learn in arithmetic can be displayed. This makes the Cuisenaire

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material one of the first that can be used year after year for different purposes, and hence the cheapest aid to date.

Because the rods are not marked they represent a unit, but in relation to another rod they gain different meanings, all easily recalled because the color works as a pointer. Thus compared to the white one the yellow is 5 and the tan 8. But end to end yellow and tan give a length equal to the orange and light green, whose values in terms of the white are 10 and 3. The equal lengths of these two pairs we call thirteen, and write 13, and thus we have 10 + 3 = 3 + 10 = 8 + 5 = 5 + 8 = 13: and so with any pair or triplet of rods.

But why compare the rods with the white one? Comparison can be made with any other. This yields fractions and the same ideas can be developed as when the white was used. In this way the difficulties contained in thinking of whole numbers at one stage using one material, and of fractions later using other materials, are eliminated.

In my writings readers can find developed in detail all the ideas that serve to show that counting has been superseded; that products can be learnt without using tables; that subtraction of any two numbers can be performed without recourse to borrowing; that all processes of arithmetic can be learnt at the same time; that fractions are not more difficult than whole numbers and can be used from the start; that a place exists for each aspect of the activities involved in learning and that children can correct themselves and reach a much higher level of mastery than is considered usual; that quick pupils advance

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further than others without being held back by the slow learners and find many more meaningful challenges; that slow pupils gain confidence, experience, and hence speed, because they go at their own pace, and that their speed is an increasing function of experience; finally, that the application of mathematics to the real world seems natural to those who have developed within themselves the structure to apply.

In working with children of many countries and of all ages, I have been able to show that many of the conceptions current among educational psychologists and teachers are substantiated only if children are taught as we were 50 years ago. The most important finding is that pupils can learn much more mathematics of a much higher level of abstraction and much faster than hitherto. Without forcing them they can now, on the whole, reach at seven years of age a level of mathematical experience equivalent to that of eleven-year-olds, and at eleven or twelve the level reached in the higher forms in secondary education. What was needed was to start from the child rather than from existing mathematics and the preconceived ideas of adults.

In an epoch where more scientists and mathematicians are needed, it seems reasonable to expect that teachers will give a suggestion received from one of their colleagues a chance.

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This article was written for a Sunday magazine in 1957 in the U.S.A. but only appeared as a mimeographed sheet in 1960 in New York.

Now that so many schools of Great Britain have had time to test the foundation on which the methods of teaching arithmetic we have advocated are based, it would seem appropriate that not only teachers but also administrators should be made aware of the hopeful news.

The traditional approach to mathematics in the primary school stressed counting as the basis of all arithmetic processes.

Counting in ones is so important in this context that fingers are still used throughout life in a great many cases. Not content with this basis however, teachers soon try to replace it with the memorization of addition of groups, the so-called numberbonds. (For example, 7 + 5 = 12 is carried out, rather than 7 + 5 = (((((7 + 1) + 1) + 1) + 1) + 1) = 12.) Here, teachers instill, in the first instance, habits which they soon after wish to eliminate.

This approach is repeated when dealing with multiplication: the

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pupils are taught the multiplication tables, but shortly after are required to know the individual products. Similarly with fractions, and vulgar fractions in particular, habits are formed which must frequently be unlearnt later. It is scarcely surprising that our pupils are confused and lose confidence; little wonder they ask what the teachers mean by this or that when the conditioning which has once been established ceases to be of use; it is only to be expected that they no longer understand what all this is about.

In the case of subtraction, teachers are divided among themselves, and the methods in use, on the one hand require long practice before they can be mastered, and on the other hand often differ from school to school; thus, when a child transfers from one school to another a new method must be learnt and an old one forgotten. First, our children pay a high price for our ignorance, later the whole of society suffers since so many of those taught prefer to abandon the study of science as a result of the lack of a good mathematical grounding. All this can now be radically changed.

Indeed we can, by making use of the Cuisenaire material, provide a different series of experiences as a basis to arithmetic learning. We can form number-bonds from the outset, making use of counting only where it is useful and proper, as when we need to know the cardinal number of a collection (how many bottles we need, for example, or how many stamps or marbles I have, and so on). These number-bonds are formed in such a way that, for any one number (represented by a length of colored rods), the children can simultaneously find first, the complementary numbers (5 + 4 = 3 + 6 = 2 + 7 = 9, for 126

14 Mathematics for All

example); secondly, those which when repeated form the numbers, that is, the factors (three threes are nine), and thirdly, can see that division is a new reading of a multiplication, followed by an addition. If 11 is understood as 2 Ă&#x2014; 5 + 1, how many fives are there in 11? or if 3 Ă&#x2014; 4 + 1 = 13, how many fours are there in 13? By moving from one set of decompositions of any one length to that of any other, children become acquainted with numbers and realize what makes them different from each other; this understanding however, eludes them when 7 and 9 are only either written signs, or sounds uttered, for beads in heaps.

This fundamental difference between the new and the traditional approach can be best seen when dealing with processes suffering from the initial assumption that formalization must precede understanding. Here counting imposes a units-tens-hundreds written approach and consequently subtraction becomes a nightmare should the number to be subtracted have a digit higher than the digit in the corresponding column of the other number. With the Cuisenaire material, this problem simply does not arise. We can achieve what we want, by-passing such obstacles.

The reader is referred to more technical publications* in which all the items to be learnt are studied in detail. Here we are concerned with the improvement that may result from

* Mathematics with Numbers in Color; A Teacherâ&#x20AC;&#x2122;s Introduction to the Cuisenaire-Gattegno Method of Teaching Arithmetic; Now Johnny Can Do Arithmetic; Modern Mathematics with Numbers in Color.

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confidence, from competent handling of mathematical problems from the first years of schooling, and from the possibility of using the time thus saved for introducing more mathematics into the primary school syllabus. It is this last point which is perhaps of the greatest significance when we consider that, with these new methods, we need no longer fear that only an elite will muster the important tools of mathematics, or when we think of the continuous expansion of knowledge now possible. The opening of mathematics to all was formerly considered possible only with a watered-down syllabus. Now we can offer an expanded curriculum to be undertaken from the primary levels.

Most of the classical geometryâ&#x20AC;&#x201D;naturally, approached differentlyâ&#x20AC;&#x201D;can be retained but as part of primary school mathematics syllabus. The stress on proving theorems as occurs in Euclidian geometry can be dispensed with without loss to anyone and can be replaced by the acquisition of spatial experience to an even greater extent than that advocated in the Primary School report of the Mathematical Association (London, Bell & Co., 1955). In the last few years, experiments in teaching have shown that an extensive body of knowledge, concerning properties of sets and of spatial relationships, can be presented to young children, and that they have no difficulty in becoming familiar with them and in using the correct language when considering them. It is necessary only to abandon the view that geometry is a mental discipline of a rigorous (i.e. rigid) character.

This transformation can take place through adopting teaching aids, or rather learning aids, of a more dynamic nature than textbooks and by letting the pupils see, manipulate, create and 128

14 Mathematics for All

take apart situations that are both concrete and schematic.

Among these aids geo-boards are the most readily accessible to the understanding of the pupils. But models, films and filmstrips also have a large role to play and will become increasingly interesting and varied as teachers develop their use.

The Association for Teaching Aids in Mathematics in this country devotes its energies to making, testing and evaluating these aids and to disseminating information about them, and is helping more and more in the work of making mathematics enjoyable, useful and accessible to all.

Teachers in all fields can obtain assistance if they wish to improve their methods. Seminars lasting a weekend or more can be organized at which they can see these methods at work and discuss them both with competent leaders and among themselves, and can obtain further information or tuition when needed. In the last few years, teachers in schools of all types have found help and stimulation in such seminars.

Since teaching mathematics to all presents a technical challenge to the teaching profession, the answer must be found in the field of method and equipment. Much of the solution is already available. The task now is to make teachers aware of what already exists, make them want to experiment, and when convinced, change their attitude and join the members of the profession who accept the challenge the needs of the future put to this generation. To say that mathematics is for all (meaning not merely the most trivial arithmetic) is no longer a utopian dream. It is indeed demonstrably true that we have before us a period of vast extension of scientific teaching in both primary and secondary schools that will alter past ideas of 129

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educationalists concerning the capacity of our children to learn and think about the real universe. A period lies ahead where there will be great scope for creativeness which will bring teachers its own encouragements and rewards.

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15 Note for Administrators

This article was written for Education in November 1956, but was not, in fact, published.

In writing for the general public we must be brief and to the point and fully aware of the real challenge presented by large classes, shortage of teachers and other difficulties.

Mathematics teaching is a question sui generis. So many people attempt to study mathematics and so often so little is grasped that the learner feels this to be due to some lack in himself. In no other subject is this felt to the same extent.

It is my contention that everyone, or almost everyone, can understand mathematics and gain a certain mastery of it. This is a matter of considerable interest to administrators in several respects: firstly, because the educational system is required to provide an ever-increasing number of technically-minded citizens, and failure in mathematics, the key to the sciences and technology, causes many to pursue arts courses instead; secondly, because what is a small part of the syllabus, badly 131

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mastered by the teacher, becomes a heavy burden for pupils in both primary and secondary modern schools, a burden which makes them less efficient in their other school subjects; thirdly, because schools must accept as teachers those who are less well qualified, since they are in competition with employers who can offer better-paid posts, with the consequent continual drop in the quality of teaching; and finally, because we have here not something which constitutes a greater strain on the system but a real hope for the future.

On what is my contention based? Firstly on extensive experimentation in this and other countries with extremely diverse groups of pupils, from special classes of blind, deaf and backward children to ordinary infant, junior, secondary modern and grammar-school classes. Secondly on the fact that simultaneously as conditions in the teaching of mathematics were deteriorating I chanced upon the means of altering in a very short time the whole approach to this teaching. Whereas method has hitherto been ignored as unimportant in the campaign for the solution of the challenges of education, there are many today who know that methods exist which, when properly carried out, can transform the situation.

Administrators should know that if they are prepared to tackle the problem from this angle they can meet the demands of an expanding economy, of an increasing population, and of a wider system of education. If every learner is brought to grasp the working of his own mind in the process of bringing mathematics into reality, as must be the case in an electronic-nuclear age, all the challenges will have been met. Then everyone, and especially those who choose to become teachers, will possess that 132

15 Note for Administrators

knowledge which can make others grasp what mathematics is and what its applications are.

This thesis—that we can, in five years, with a carefully planned attack, transform the teaching situation, is developed in a booklet shortly to be published in two sections.* Here I can inform the reader that, thanks to Cuisenaire’s invention of Numbers in Color and the developments that have followed in many schools in Great Britain and abroad, we now have proof that many preconceived ideas about the “mathematicallyminded child” are a hindrance to the aim of a mathematical education for all children. It is, in fact, quite easy to prove to anyone of good-will that, by using Numbers in Color or materials created for the purpose of making mathematical thinking functional, any ordinary child, and any teacher not cramped by prejudice, can reach a level of mathematical performance hitherto unsuspected by students of education.

This is a technical solution to a technical problem of teaching and requires only that it be given a trial. The urgency of the situation demands that this be done.

* Teaching Mathematics in an Expanding Economy, Reading. Section I (Primary) 1956.

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16 Multivalent Materials

This article was first published in Le Matériel pour l’Enseignement des Mathématiques, Delachaux et Niestlé, Neuchâtel & Paris, 1958.

1 Inventors of mathematics teaching aids have a tendency to devise objects to serve either one, or several, purposes. For example, a plaster cone, already cut in accordance with the various conic sections, is designed to show only the shapes met in conics. But a strip of paper can, according to what is done to it, become a cylinder or a Moebius surface; Meccano parts, according to how they are put together, provide various compositions of transformations.

Materials which are deliberately designed for various purposes I shall call multivalent, and univalent those equally deliberately designed for one single purpose only.

Teachers are divided for and against each of these types of materials, and I must straightaway say that, personally, while favoring the multivalent type, I recognize that there is a place for 135

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the other, and that it is sometimes better to use a univalent material if this makes achieving the desired aim easier. But this is a question of values and thus better left aside in a chapter intended to be practical.

2 In my chapter on perception,† I underlined the importance of dynamism both in thought and for an adequate apprenticeship in mathematical thinking. In my chapter on films, †† the idea of the dynamic pattern was developed. Readers will therefore already know why I favor materials which make use of movement and which lead to one’s becoming aware of relationships. But I must show how, in my search for materials, I made use of this polarization towards multivalence, and it seems to me that if this practice were more universally adopted, we should soon be provided with a multiplicity of aids which would facilitate the task of the mathematics teacher.

3 The first example I shall describe is that of the geoboards, which are aids to help one to become aware of geometrical relationships. They can be described in the following way: on a board a given pattern of lines is traced, and nails hammered in at a number of selected points thus providing various supports on which differently colored rubber bands can be stretched. The patterns I have used are those for the regular dodecagon, the decagon and the octagon; rectangular lattices of 9, 16, 25, 49, 81 and 121 nails; in addition a double hexagon has been used and a number of others besides from time to time.

† Reproduced in Volume 2, Part III, Chapter 4 of this work. †† Reproduced in Volume 2, Part IV, Chapter 3 of this work.

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With the aid of these few geo-boards, I believe that most of the theorems of Euclidian plane geometry can be taught. But first their use needs some explanation. In the same way that nobody objects to using the blackboard for drawing figures, it seems to me that no-one will be opposed to using a geo-board, which is such that a figure appears when an elastic band is stretched over a set of pins. But, in this, the geo-board already has certain advantages over the blackboard. Polygonal figures are accurately and clearly formed, and do not depend on the skill of the teacher; as the geo-board is relatively small, it can be turned round and the figures shown in any desired position, thus illustrating that the properties displayed are invariant with respect to displacements. Anyone using the geo-boards is advised to adopt the rule of turning the board round while showing what is formed on it to allow each figure to be considered from different angles.

Another way of displaying this invariance results from the fact that a certain choice of a given number of points (three for example) can be made in several ways. If the elastic band used for each choice is of a different color, figures are obtained (here triangles) which, by definition, are congruent, in the sense that there is no reason for preferring any one set of these points to another.* If each pupil has his own geo-boards, he can throw himself into the activity of investigating and discover first the entire range of

* It seems to me that more use could be made of this idea of indifference of choices, which would appear to be a basis for mathematical proofs.

139 137

Part V Elementary Mathematics

figures which can be formed with the aid of the rubber bands stretched over the pins; then those figures that are congruent, counting them, proving that he has found the right number by discovering a rational principle which allows this deduction to be made (rotation, symmetry and translation, and combinations of these); finally, he can discover those which are similar. He can set out to find all the possible quadrilaterals in the lattice, to distinguish the convex from the concave, the regular from the irregular, to classify from the general to the particular and to the still more particular, or from the most special to the most general, discovering that a parallelogram is in two ways a trapezium, or that a square is a rhombic rectangle or a rightangled rhombus. If the diagonals of quadrilaterals are considered it can be seen that those of the trapezium (if this is not isosceles) possess no more immediate observable properties than those of the general quadrilateral, whilst properties increase in number as one specializes more and more (a precious mathematical experience). The square, although the most familiar of all quadrilaterals, is also the most special, and its diagonals possess the properties which belong separately to the diagonals of other quadrilaterals.

With the geo-boards no drawing is necessary; the elastic bands are simply moved from one pin to another, additional ones may be used if desired, and what is seen is immediately grasped. The ease with which the figures can be changed and different constructions used makes the tool eminently flexible.

The geo-boards for the regular polygons can serve to illustrate the way in which theorems are grouped. Here is an account of what may be done with, for example, the decagon and the 138

16 Multivalent Materials

dodecagon boards. By stretching the elastic band round the external pins a regular convex polygon (of 10 or 12 sides respectively) appears. By starting at one point and joining every other one, two regular convex polygons with 5, or 6, sides can be obtained, being half the number of sides of the decagon or dodecagon. If every third point is joined, in the first case a stellated decagon is formed, and in the second case a square. If every fourth point is joined, a stellated pentagon appears in one case, and an equilateral triangle in the other; by joining the fifth points we obtain a diameter and a stellated dodecagon respectively. In the case of the dodecagon, a diameter is obtained by joining every sixth point. This study is interesting per se, and if it is completed by examining an octagon, already certain properties of regular polygons with an even number of sides can be seen. In particular, on comparing the convex and the stellated polygons (as, for example, on the decagon board) it is found that s52 + 4a52 = 4a105 + s102 (where s stands for side of the convex polygon, and a for its apothem) and that the apothem of the one figure, pentagonal or dodecagonal, is half the side of the other figure (decagonal or pentagonal), such that a'5 =

s10

a'10 =

s5

a5 =

s'10

a10 =

s'5

(where the dash indicates the stellated polygon).

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Part V Elementary Mathematics

Starting again with the same geo-boards, the angles at the circumference and the angles at the centre can be studied. This use of the geo-boards is very much appreciated by the pupils and I can recommend it to those who wish to assess the value of this apparatus.

Starting from a diameter, a right angle can be formed by stretching the elastic band over any one of the points on the circumference. By placing a rubber band from this point to the centre pin, two isosceles triangles are formed and it can immediately be stated that the angle inscribed in a semicircle is a right angle. Leaving the elastic band around one of the pins of the diameter and also around the previously chosen point on the circumference, the band can now be moved in such a way that when it is stretched over various other points on the circumference, what had formerly been a diameter now becomes two radii. When the figure is examined it can be seen that the angle at the circumference can assume values greater or smaller than a right angle and that the angle at the centre, which varies in the same movement, can be compared to it by producing the radius which had previously been inserted as far as the opposite point on the same diameter. The same process as in the case above (the consideration of the two isosceles triangles) reveals the relationship linking the angle at the centre to the inscribed angle. In particular, it can be seen that the right angle is half the angle formed by the straight line at the centre, a fact which had not been previously noticed.

The ease with which the figure can be changed, revealing that the various shapes of particular figures are only aspects of one

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16 Multivalent Materials

same relationship, makes us realize that the facts displayed can be expressed in various ways: 1

the angle at the centre is double the corresponding angle at the circumference;

2 the angles at the circumference whose sides pass through the same points on the circumference are equal, being both half the same angle at the centre; 3 all angles at the circumference whose sides pass through the extremities of the same diameter are right angles; 4 all angles at the circumference whose sides pass through the extremities of equal chords are equal or supplementary; 5 the property (a) does not depend on the size of the angle at the centre or of the angle at the circumference, provided that the first is less than four right angles and the second less than two right angles; 6 the property (c') is the same as saying that the opposite angles of a cyclic quadrilateral are supplementary if it is convex and equal if it is not; 7 squares, rectangles and isosceles trapezia are cyclic quadrilaterals, while parallelograms and ordinary trapezia are not.

With the rectangular geo-boards, one can set questions on areas, lengths and sizes of angles which can be easily solved, but which nevertheless amount to considerable mathematical experience.

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Some of these questions are difficult enough to demand several hoursâ&#x20AC;&#x2122; work from a good student of 17 or 18 years of age.

It is because the same apparatus and the same actions allow the setting of a variety of problems at all levels that I call the geoboards multivalent materials. Lessons in which they are used are most successful, and are not easily forgotten.

There are three types of lessons with the geo-boards: (1) the teacher can use them in place of the blackboard; (2) the children can use them to find out what they can by themselves about any particular situation given to them; (3) they can be used systematically to exhaust all that they contain. Up to now they have been used for the first two purposes, and the results, which have been very promising, augur in favor of their use at all levels of teaching. But, in my opinion, it is the third type which corresponds to my deepest pedagogical intention. In fact, the geo-boards are capable of offering definite situations, sufficiently schematized to be simple, but fertile, that is to say easily ramified and conducive to new points of view. In this way they serve in the education of the mathematician in each of our pupils.

4 The second example to be mentioned is the Cuisenaire material or â&#x20AC;&#x153;Numbers in Colorâ&#x20AC;?, which I have studied at length in a number of publications.

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16 Multivalent Materials

In addition to the rods,* Cuisenaire has devised various cardboard materials, (wallchart, lotto and cards) to supplement the rods in certain studies (multiplication, fractions, factors).

Clearly the rods can be used for building games. However, from the point of view of the apprenticeship to mathematics, here is a brief list of the uses to which they can be put. â&#x20AC;˘

Study of Whole Numbers and Operations Involving whole numbers

If a length is formed at random by placing rods end to end, the number of white rods contained in that length will be the cardinal number to associate with it. If different rods are placed alongside this length, taking care to form an equal length, a new â&#x20AC;&#x153;decompositionâ&#x20AC;? of that number is obtained. If as many decompositions as possible of this length are made and examined, we see that the four operations are contained in them. Addition is the result of placing rods end to end.

Subtraction is involved when one becomes aware of what is missing from one length in order to make it up to a given length, and in the realization that the parts which form the whole are complementary. Multiplication appears as repeated addition, and division as the realization that the repeated subtraction of a certain length can always be completed by another length smaller than the one used in the repetition in order to form the given length; this is simultaneously division as sharing, or

* The description of which is omitted here.

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Part V Elementary Mathematics

partition, (for example, giving the same number of objects to so many children) and division as quotition (how many times so many objects are contained in a given number of objects). Note that multiplication and division with remainder are similar in this set of decompositions, the last of which can be considered as being a repetition that “fails”, while the first is “successful”.

For me, the reversal of multiplication is better expressed by fractions than by division. Looking at iterations we see the factors of the number in question: for the rod used in the successful repetition represents one factor, and the number of times it is used gives the other.

The fact that all the operations can be read in the set of decompositions would alone make the Cuisenaire material a multivalent one.

•

Study of fractions and operations with fractions

Measuring one rod with the help of another generates ordered pairs expressing this measure. In general, the result is not a whole number. Let us take (a, b), the pair of rods where b measures a. We find that (a, b) and (a', b') are equivalent pairs if a'= ka, and b'= kb, where k is a positive whole number. Thus we are able to replace each fraction (a, b) with the set of its equivalents (ka, kb). We call these pairs equivalent-A. Now we are using the rods in pairs and we can see how this material allows a modern point of view to be introduced into the teaching of fractions at the primary school levels.

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16 Multivalent Materials

The notation of addition, subtraction and multiplication developed on integers is used with fractions even though it has another meaning. The manipulation of rods and the mental work associated with what the color patterns appear to contain from this point of view allows us to understand how the operations in mathematics are extended. No material created ad hoc possesses the qualities of the Cuisenaire material in this direction.

In this way, placing rods end to end was the definition of the addition of whole numbers. Now we can see that if we begin with a pair of rods (a, b), since a can be equal to c + d, we can have (a, b) = (c + d, b), in which we read that c and d are measured by the rod b, as was a. By convention, we shall write (c + d, b) = (c, b) + (d, b), in which the duplication of rod b appears in order to indicate that b is the unit of measure; the expression also shows that the operation bears on the terms measured, which are rods placed end to end, and here represented by addition. Conversely, (a, b) + (c, d) has no meaning unless, through the intermediary of equivalents, these two fractions are reduced to fractions having a same unit of measure. However (a, b) + (c, d) = (ka, kb) + (hc, hd) is always true whatever k and h and if, in addition, kb = hd, then from the above convention, this is also equal to (ka + hc, kb).

Subtraction follows the same pattern.

As regards multiplication, the problem can be presented in the following manner.

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Having noticed that, if two pairs of fractions (a, b), (b, c) have the middle terms equal, (a, b) of (b, c) is the direct relationship (a, c) and having seen that the series of intermediate terms can be as long as we like, that is, that (a, b) of (b, c) of (c, d) of (d, e) of (e, f) is equal to (a, f), what then does (a, b) of (c, d) mean?

We shall call equivalence-M the equivalence (a, b) of (b, c) = (a, c); in one direction it will be a contraction and in the other an expansion.

Clearly, if [(a, b) of (c, d)] = [(ka, kb) of (hc, hd)], and if k and h are chosen in such a way that kb=hc, we shall have:

(a, b) of (c, d) = (ka, kb) of (kb, hd) = (ka, hd) with h =

or

(a, b) of (c, d) = (ka,

)=(

ac,

bd) = (ac, bd).

In this result it can be seen that, among the fractions equal to (a, b) of (c, d) there is to be found one in which the numerator is the product of the numerators a and c, and the denominator the product of the denominators b and d. From this fact we assume the authority to denote the operator “of” by the sign × between two fractions, even though it may be a complicated operation in this instance and not simply repeated addition.

Division of fractions being the reverse of multiplication we have, by definition, (a, b) ÷ (c, d) = (A, B) which is equivalent to (a, b)

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16 Multivalent Materials

= (A, B) Ă&#x2014; (c, d). Hence ka = Ac and kb = Bd or A =

and B =

,

Therefore, (a, b) á (c, d) = (

,

)=(

,

) = (ad,

bc). This can be interpreted as (a, b) of (d, c), or the product of the first fraction by the second fraction inverted.

This formal manner of presentation is readily accessible to young pupils of eight years of age who will master its content if the order given in my Book IV* is followed. â&#x20AC;˘

Solutions of simultaneous equations

I shall take one example from among many of the uses to which the Cuisenaire rods can be put.

Let us begin with the orange rod, for example. It is worth a black followed by a green; but the black is worth a green and a pink, and therefore, the orange is worth a pink and two greens.

Moreover, the pink is the difference between the black and the green, while the orange is their sum. Thus the question can be asked: if one is given the orange and the pink, and is told that the orange is the sum of two rods and that the pink is their

* Mathematics with Numbers in Color, Educational Solutions Worldwide Inc.

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Part V Elementary Mathematics

difference, which are the two rods? Or again, knowing the sum of, and the difference between, two lengths, find the two lengths.

Although the equations proposed are of the same type: X + Y = a and X – Y = b, the rods offer an excellent means for discovering the solution, at first in the form of 2Y = a – b, X = a – Y, then under the form 2X = a + b, Y = a – X, for the understanding of the equivalence of these solutions (which brings with it a mastery of the algebra contained in the situation); for seeing how problems proposed in arithmetic text-books may be thus schematized, when the facts given are the sum and the difference of two magnitudes.

•

Miscellaneous uses

With the help of the rods the following points can be readily illustrated.

1 If a fraction is smaller than 1, it is increased by adding the same quantity to both of its terms; if it is larger than 1 this process will diminish it.

2 If a sum is distributed in relation to another sum, as many products are formed as there are in the product of the number of terms of the sums. In particular, the square of a sum or of a difference, the cube of a sum, and so on, can be easily obtained.

3 Arithmetical progressions and their properties are very simply displayed.

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4 The theory of integral indices, and thus of geometrical progressions; logarithms as â&#x20AC;&#x153;isomorphismsâ&#x20AC;? of the structures of addition and multiplication.

5 Combinatorial calculus; some elements of the theory of groups of substitutions.

6 The calculations of surfaces and volumes of prismatic bodies; in particular, the calculation of surfaces and volumes of similar three-dimensional bodies.

In addition, the Cuisenaire material may be used in a host of other ways at different levels of teaching. I should mention in passing its use in the teaching of languages.* Conclusion That other multivalent materials exist is beyond doubt (films are an example, although restricted to the field of one question), but I wanted to limit myself to well tried materials, and the two examples chosen have been the subject of hundreds of lessoninvestigations with children in many countries and of all ages. In particular, I have seen how deaf-mutes, generally considered to be gravely handicapped, could, with the help of these materials, reach a level comparable with that of pupils gifted with speech.

* Further literature is now available published by Educational Solutions Worldwide Inc.

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Part V Elementary Mathematics

Clearly, from the economic point of view, multivalent materials are to be encouraged. However, it seems that human inventiveness is carrying us towards univalent solutions. If the problem is to understand better how a situation is organized, then obviously the mind will try to produce a model which has all the required qualities, or a film which reveals the property sought after, or an apparatus which brings about the necessary transformation.

But if the problem is the teaching of mathematics, the position taken by the seeker is different. In the desire to make arithmetic accessible to children of six years of age, for whom he knew the concrete to be the field of expression, Cuisenaire set out to present a material which already contained in its structure the basic structures of arithmetic. This is the reason why the relationship of equivalence (color or length) is its backbone, so to speak. The act of placing the rods end to end is so simple, yet an act which allows the transcendence of the set of given rods since it allows that one pass on to lengths not contained in the set. But it is also the operative base of the whole of the algebra of rationals. The structure of the field of the rational numbers being found in the set of rods, it is not surprising that these rods form the material for the teaching of arithmetic. There is no question in the syllabus which cannot be illustrated, or illuminated, with their help if it only requires a finite set. The multi-valence of this material is its greatest pedagogical and economic asset.

The geo-boards, although the area of their application is more limited, are an apparatus which commends itself both for its simplicity and for its many possible uses. If the purpose is 150

16 Multivalent Materials

education for geometrical sense as an aptitude distinct from the acquisition of knowledge, it offers excellent experience in the generation of figures, and puts forward problems of a wide range of diversity. The ease with which a perceptive field transforms itself and structures itself to reveal one or several relationships is, without doubt, the greatest contribution to the development of a geometrical sense, unfortunately so neglected in traditional teaching.

Let it be noted once again that, in the dynamic teaching which forms the background to the discussion of this chapter, I am in no way concerned with general theory. I have been thinking about teachers engaged in a dialogue with younger minds, and attempting only to put forward to them fertile mathematical situations. Whatever the ideals of the reader, I hope that my technical suggestions and my attitude, which enables pupils to find what a given situation contains, will be recognized as being based on the reality of the mathematics classroom membership made up of minds contemplating situations which the teacher or circumstances have presented.

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17 The Cuisenaire Material is Not a Structural Apparatus

This article was written for publication in South Africa in 1962 but has not appeared at the time of printing this book.

This is the first article I write especially for our South African colleagues. It was prompted by the desire to let them know that Monsieur Cuisenaire and myself are very moved to see the growing interest in our work in a country many thousand miles away from the places where we developed our approach and techniques, and one of the few places I have not yet visited.

Then Dr. B.â&#x20AC;&#x2122;s article in the Nov. â&#x20AC;&#x201C;Dec. 1961 issue of The Transvaal Educational News made me wish to add my voice to the many already heard in your country. This I shall do by putting down a few unrelated notes, one after the other, hoping that readers will take them one by one and consider them seriously, since they often deal with points rarely met in the manuals and other published materials.

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1 A set of Cuisenaire rods is much more an algebraic model than a geometric or arithmetic apparatus. Simply to see it is sufficient for one to realize that with its help we can illustrate the theory of indices up to any value whereas â&#x20AC;&#x153;cubes and flatsâ&#x20AC;? display only three and two dimensions. Of course, the many uses I have found for the rods will convince the reader still further; they can be used to introduce equations, simultaneous equations, progressions, permutations, and all that is known to be algebra and not geometry.

2 It follows that if we wish to use the rods properly we do not think of them as an apparatus for structural arithmetic as is the Stern, created to teach number only. The main characteristic of Numbers in Color is that we start with qualitative arithmetic, which is another name for algebra. Because pupils meet first the less structured and move steadily towards more and more structured entities, we can expect that they behave mathematically in an entirely different way from the way we do and from all who base their knowledge on counting. Instead of meeting operations incidentally when studying number, we find that we can transform our newly acquired operational powers into a great many mathematical entities (of which number is only one example and perhaps not the most important) by specializing the situation offered by the rods.

Indeed, children working with Numbers in Color are as if they were of another essence, having shown everywhere that they can invent mathematics, even when their I.Q. was supposedly low.

This is understandable, for a powerful means has been made available to them whereas the memorization and notational

154

17 The Cuisenaire Material is Not a Structural Apparatus

juggling of my schooling and yours are, for most people, unrelated to inner experiences.

3 Every day I hear that people, who base their objections to my work on opinion, have been proved to have preferred opinion to fact. If it is true that our pupils can do what someone says, what good will it do anyone to deny that fact? Is it not better to test the claim at once rather than turn one’s face in another direction? In less than 9 years since I met the colored rods for the first time (and 22 years after Cuisenaire started investigating his intuition) not only have some one hundred thousand teachers in 80 countries adopted the rods (and their number increases every day which goes to show that a good idea will reach people even without publicity), but also the experimental evidence in classrooms is now overwhelmingly convincing. Teachers have reached the finishing post before the official investigators because they are first moved by success in the classroom while the “Authorities” are only impressed when they see figures on paper, starting to investigate teachers’ findings only when they cannot do otherwise. The authorities are more and more perturbed by the truth of my seemingly preposterous claims in 1954; that all problems met so far in elementary mathematics learning vanish when the Cuisenaire rods are properly used. “Properly” means: the creation of operational awareness before number names are introduced. It is in the relationships expressed through the colors that we find the facility needed. Hence the great advantage of the functional colors of Cuisenaire over the beautifying colors of all other apparatus, including the Stern.

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Part V Elementary Mathematics

Indeed mathematical colored rods would be a name that would describe the rods more accurately than does their present title of numbers in color given them by Cuisenaire which seems to restrict their use and value to the introductory stages at school.

4 A number of new names in the field of education are emerging as those of new leaders who have really understood what transformation the introduction of the rods has brought to our psychology both of learning and of our pupils. People like Madeleine Goutard in Quebec, William Hull and John Holt in Cambridge (Mass.), Lore Rasmussen in Philadelphia, Alan Strain in Palo Alto, John Trivett in Seattle and Roland Genise in New Yorkâ&#x20AC;&#x201D;to quote only from North Americaâ&#x20AC;&#x201D;are accumulating the evidence needed to substantiate the new claim of psychologists, such as L.J. Bruner of Harvard and Piaget of Geneva and Paris, that children can actually do much more than has been assumed till now.

This is indeed my greatest reward so far. Children are at last taken into account. Mathematics, the subject at school that once made us be regarded as either bright or dull, is now saving our children from paying for our ignorance. It is working as a gale, sweeping away all cobwebs hung up in times of idleness with the breath of reality, and we can see at last that no child is dull but rather that dull teaching and ignorant handling of educational situations have created legions of dulled minds. That such a phenomenal change of attitude among teachers and psychologists has been made possible by a proper use of the rods would be sufficient for me to avoid classifying them as apparatus and to recognize in them the help they bring to our minds.

Indeed, there is a mystique of Numbers in Color due to the fact 156

17 The Cuisenaire Material is Not a Structural Apparatus

that it is an eye-opener as well as a consolation for all those who believed that there was a difference between being a dud and not being able to do mathematics at school. In my courses for teachers in so many countries it was human relationship with the participants which made me feel that at last we could educate people while they were acquiring the skill â&#x20AC;&#x153;to do sumsâ&#x20AC;?.

Instead of creating lopsided minds in our mathematics classes where at most only a skill is acquired and the soul often lost in the process, we can now see how our children expand intellectually under our own eyes and reach hitherto unsuspected levels of understanding. As most people are capable of this we are entering a new era in education from which we can expect a great deal.

To the children of South Africa the arrival in their country of these colored pieces of wood will be an event of importance.

157

18 Why My Books Are as They Are

The article first appeared in Cuisenaire News, No. 3, January 1963.

If selling books were my prime concern, I should offer what is most attractive to teachers.

But I am only interested in doing all I can to contribute to child education. My books will die a natural death if I am mistaken in believing I have something to offer and I should not mind as I would have learnt my error.

Since I can write about my books for pupils here and since I have seldom seen a review of them anywhere, I shall attempt to explain why they are written in the particular way they are. I do not expect thereby to make them more appealing to those whose judgment reflects their preconceived ideas, but I hope to bring the more sensitive readers to my point of view.

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Part V Elementary Mathematics

Let me first state a point of fact. My first three books were originally published by Heinemann, who decided the style and size of type face, the general appearance, and the quality of paper and covers. The first reprints had a different cover from the original, this being the result of my only interference with the publishers’ wishes. But the content is all my own and I take full responsibility for it, in whatever form the books appear.

There are ten books in the English series.† They were written between August 1956 and the end of 1957, with the exception of Book 4, which was completed in January 1960. Many people know that from 1953 to 1956 I refused to follow any suggestion from teachers that I should write text-books. My reasons were 1 that I wanted the teachers to use the rods according to their lights and not to mine; 2 that, just as I did not want to interfere with the learning process of children, so I also did not wish to interfere with the teachers’ freedom of work.

But so many requests could not all have been wrong and I struggled within myself to find a proper solution to these conflicting factors.

The solution came in Spain, in August 1956, while running a course for teachers in which I gave lessons for most of the time.

†

The revised edition in seven volumes is to appear in 1963.

160

18 Why My Books Are as They Are

My idea for a book was that of a log, recording a set of lessons with actual children. A set of questions—what better could be put into a book? In this way the experimental basis of the approach would be preserved and, as there was no need to give answers to the questions, both teachers and pupils would be free to investigate. My main objections to writing books had been overcome.

Any student of mathematics teaching who looks at textbooks will find that they are what they are partly because of the theory of learning and teaching which underlies the content and partly because of the kind of material offered. Coming to mine, I hoped that everybody would know before looking at them that they were to be used in conjunction with Cuisenaire’s rods and his cardboard material. The approach of question and answer in a dialogue about situations formed by and with the rods is evident in Books 1–6. I only depart from it from Book 7 onwards but return to it whenever possible.

The first question to be answered concerns whether the books for pupils are companions to the rods. Without the rods they are incomprehensible. They are written in a language foreign to anyone who does not use the dictionary of manipulations with rods found in each section. But they become child’s play when used in conjunction with the rods, which is precisely what I hoped to achieve.

In Books 1–3 there are no diagrams (except for the clock in Book 3). Anyone who inserts diagrams has, in my opinion

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Part V Elementary Mathematics

1

completely misunderstood the components of learning with the rods, which is a mathematization of actions;

2 mixed an old-fashioned and no longer justified approach, based on images suggested by illustrations, with a dynamic approach based on actions with objects, thus reducing the efficiency of both; 3 slowed down the learning process by spending time on irrelevant work; 4 fostered habits of thought which are hybrid, thus undermining the integrity of the learnerâ&#x20AC;&#x2122;s mind.

It is easy to find that authors of textbooks capable of tapping the real fund of psychological experience of learners never destroy the value of their teaching based on actions by referring to diagrams which neither add meaning to what is asked nor serve any purpose other than that of satisfying prejudices in teachers who have been taught through the eye or the ear and not through the hands.

Our elementary mathematical ideas are actually formed by recognizing what effect our actions have on situations and by seeing that some factors or attributes are irrelevant to some viewpoints while others are relevant. Moreover, mathematical activity lies in the awareness of the dynamics of relationships, and my texts are a profession of having undertaken such studies.

Whoever reads them in order to criticize them genuinely and does not reject them on appearance will find that every page is carefully considered. The content is dictated by an analysis of the activities underlying the work. If the results can be achieved 162

18 Why My Books Are as They Are

by manipulation, observation, and verbalization or notation, then this is all that is asked for from the reader. If the language and the notation must be learnt before one becomes articulate in thinking of a situation, then the exercises are essentially linguistic. If one is to become quick at calculating then one must practice equivalences and operations per se. Four stages in the learning process have been observed and are taken into careful consideration from the first book to the last. These stages are as follows: 1

Contact with the situations in order to become familiar with them, noting the relevant and the irrelevant.

2 Analysis of the components of the situation in question so that their various forms and ways of presentation may be recognized.

3 Mastery which has been achieved when the activity is accompanied by joy and the realization that all forms are potentially available under any one actual form. Mastery is the aim, but this really is only realized when the fourth stage is reached.

4 Application of the knowledge acquired to new situations or symbolic forms.

Although all this may appear to be highly sophisticated and not obviously present in these books, it is put there explicitly as the stamp of my mind and as it has been generated in my experience. As my teaching has been very successful, having been widely adopted by others, I feel that the technique used in the writing of these text books, which follows the development of ideas in the young, cannot be really wrong.

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It is sometimes believed that watering down food makes it more acceptable. Rather it is made less nourishing and perhaps tasteless. I should like to enlist the co-operation of readers in order to prove to the profession at large that the original work, which is now copied so freely, has merits which it would be a pity to lose. At least I can give guarantees, which are not often associated with imitators, that I gleaned my knowledge from hundreds of lessons in schools during years of research and that it was continually checked on larger and larger populations. My writings are simultaneously studied by French, Spanish, Italian, German, Greek, Portuguese, Hebrew, Amharic and English speaking children. From everywhere I hear that there are children and teachers eager to use my texts, but that most teachers find them difficult. Can they really be hard for teachers if learners between the ages of 5 and 11 find them exciting and useful? I, for one, cannot believe it. It must be that these teachers do not give them a fair trial because their appearance is not what they are accustomed to.

One last question to the reader. Does he think that I, as an author, should forget my experience and the contribution I can make to education, and publish my books so as to attract the greatest possible number?

164

19 The Place of Color in Mathematics Learning

This article first appeared in Mathematics Teaching, No. 23, Summer 1963.

Mathematicians are supposed to be concerned mainly with abstractions, though it is difficult to say what these are. Some mathematicians even talk of different levels of abstractions and suggest that classes are more abstract than individuals, and classes of classes more abstract than classes, the most abstract being the one about which almost nothing is assumed except that it exists and that it has elements between which certain relationships are to be found. Classes naturally are not only considered in mathematics; they occur in all intellectual fields of study from grammar to law, passing through the natural sciences. Analogy is a highly abstract activity made use of by all who learn from the experience of others. Model making and the use of plans and maps also assume that one finds in one reality what is suggested by another so long as one is concerned with relationships and not with the actual thing.

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Numbers become a mathematical reality when they are defined as labels for classes of equivalence of sets. But numbers have many attributes when considered as objects of the mind, and the study of these attributes is the difficult discipline called theory of numbers, a field open to amateurs only where trivial results are concerned. The behavior of numbers remains a mystery fascinating and bewildering mathematicians who are full of respect, even awe, for their individuality.

Amateurs and school-children have still to learn to meet mystery as such and with respect. Our school curriculum does not help in that it aims at results and skills rather than at educating insights into difficult and complex fields.

A few years ago, when I was asked to visit Cuisenaireâ&#x20AC;&#x2122;s school for the first time, I replied that color has nothing to do with numbers. I still think in that way, my reasons being that colors are one of three things. They are either (1) labels given to frequencies of electromagnetic vibrations, and whose description requires only the name of whole numbers or most rudimentary properties; or (2) impressions on the retina that generate constant appreciation by our minds and are made uniform in society through language; or (3) pigments used by artists and other professionals in specific ways to procure impressions upon the retina. In all these cases colors have attributes that exclude their application to numbers as far as their behavior is concerned, unless it is wanted simply to give a rudimentary analogue for rudimentary properties of some very special numbers.

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19 The Place of Color in Mathematics Learning

Yet, in spite of the deep conviction that number has nothing to do with color and conversely, I am responsible for the propagation all over the world of â&#x20AC;&#x2DC;Numbers in Colorâ&#x20AC;&#x2122;. This phrase is, of course, both a joke and a trade mark. As a trade mark it refers to the article produced by Cuisenaire of Thuin in 1952 for use by others. As a joke it tells me, and anyone who may have shared my views, that our right to reject color only applied within mathematics and not to the learning of mathematics. Color may have a lot to do with learning mathematics, as indeed I found as early as 1953 when I started teaching infants with the Cuisenaire rods. What is the significance of this?

Euclid and Descartes have accustomed us to considering a set of segments on straight lines as isomorphic to the set of whole numbers, or to the set of rational numbers (or to the set of real numbers later on). A set of lengths made of wood can serve as a model of a set of lengths on a line. Most readers know how actions upon the set of rods can provide insights into the behavior of lengths and therefore of numbers.

It being understood that the mathematics we shall obtain from a set of rods is derived from their lengths we shall have no worry about the content of our course, which will be essentially mathematical. The questions that remain are: 1

What can we gain if we have a set of rods instead of segments on a line?

2 What shall we lose?

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3 Can we generate an optimal model using some attribute other than length to represent it?

What we can gain is that manipulation generates situations which, of course, cannot be done actually on the line. Moreover, we know today that the structures of modern mathematics are more easily found in the structures of our awareness of relationships than anywhere else, thus permitting us to generate the notions rather than impose them.

What we can lose it is difficult to say, particularly for me, having attempted over a period of ten years to draw out of the set of rods as much as I could and having written so much about them.

It may be that there is a loss when we pass from the set of rationals to the set of reals, a passage accomplished so easily on the line but impossible with a set of rods.

Though Cuisenaire was not well acquainted with modern mathematics, he was alone in proposing that we use color in order to generate an optimal model for a set of lengths endowed simultaneously with several of the structures that permit a large number of mathematical properties to be demonstrated to anyone who can see and/or feel. His was the first use of color for some functional purpose, though color had often appeared on beads, counters and rods before.

Note that color, for Cuisenaire, is not a substitute for number, but that color relationships can be made to appear isomorphic to some relations. Equal lengths display the same colors and conversely. Related colors are used to show some of the numerous relationships possible between the lengths of the 168

19 The Place of Color in Mathematics Learning

rods. Cuisenaire was lucky enough to choose colors which made his set capable of acting as a model for the set of rational numbers. He did not restrict himself to one relationship, but involved his lengths in a number of series (arithmetic, geometric, harmonic), using his idea of families and affinity. His choice of doubling as one of the principles for coloring, and of darkening as an indication of increase in length within one family, are such lucky intuitions that they convey without words to anyone playing with the rods information which can, on questioning, be articulated even by five-year-olds. The reasons, as I know them, are that doubling is indeed a deep physiological experience, and that the use of one or two hands (informal tearing of paper in halves for example) leads naturally to the geometric series a Ă&#x2014; 2n, a being the starting point and n the number of times we apply the process. For Cuisenaire, darkening was linked with musical theory and organ pipes, but a more immediate experience is found in the affective tone of voices or oneâ&#x20AC;&#x2122;s own voice: deeper, lower voices convey to us spontaneously an association with darker tones. This can be tried out at once by any reader. Another test can be the spontaneous association of various colors while painting and listening to music of different tones. This very old experience is mentioned in the literature of the past, and is seen in the colors used in stained glass windows in churches and mosques.

Cuisenaire will only agree to indicate how he finally settled the question of the colors of his rods when he is sure that he is talking to a very sympathetic listener. Nowhere can his views be found stated in full. My own attempts at explaining his intention have apparently been accepted by him as correct, but I am not sure that it is the expression of what actually happened. No-one 169

Part V Elementary Mathematics

to this day has suggested a coloring that has the power of evocation of Cuisenaire’s. Indeed it seems that for learning mathematics uncolored rods would be as good as colored ones.

Over a century ago some were already in use; they were reinvented a number of times but lost any ground which they may have gained. What Cuisenaire did through his individual coloring of his rods (or through these same rods uncolored) can be summed up in the words I used in the preface to the first edition of ‘Numbers in Color’ in 1954: he made it possible to teach mathematics as a set of relationships, and also—as I was able to say only in 1957—showed us that algebra is easier and more primitive than arithmetic.

Now, of course, all this is widely known and proved again and again every day in schools throughout the world. It has been my privilege to be the first to make it explicit, but had it not been true I could never have established it. Something like the Cuisenaire rods was necessary in order to discover that, and in this way the use of color in the teaching of mathematics is indirectly justified, at least as Cuisenaire proposed using it.

It seems legitimate to require that any author of an alternative proposal show at least one new use of other color schemes to prove that there is some advantage to be gained from adopting the new proposal. As far as I know, none have been able to do so up to the time of writing this article. But this does not mean that the field is wholly surveyed and charted. Let serious people consider the matter seriously, and whenever a real improvement is proposed it will certainly be taken up as the Cuisenaire method has been in the last ten years. Because of what has been brought to children to help the better learning of mathematics, 170

19 The Place of Color in Mathematics Learning

and to teachers to be clearer about issues involved, we can say today that it was a very happy moment for us all when a primary school teacher in Thuin resorted to color to teach number.

171

20 Discovering Cuisenaire

This article first appeared in Cuisenaire News, No. 3, January 1963.

Although I have written a great deal about the teaching of mathematics using Cuisenaire rods, I have rarely discussed the significance of my visit to Thuin in April 1953. This article is a further tribute to Cuisenaire and will be more autobiographical than my other writings.

I remember it very clearly: a telephone rang during a dinnerparty and someone said: “As you have nothing to do tomorrow morning, would you like to visit a school where a teacher teaches number through color?” Stupidly, I immediately replied: “But number has nothing to do with color”. However, I could not refuse as I was the guest of the province and the teacher in question was only ten miles from my hotel. It was remarkable to find that no-one I knew in that area concerned with the teaching of mathematics had ever heard of this teacher and his experiments. The following morning, accompanied by two Belgian teachers, I took the train to Thuin where an elderly

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gentleman, wearing many decorations, as is customary in some European countries, met us at the station. He was Georges Cuisenaire, a retired Director of Education for Thuin. He was pleased to see us and took us to a school where an ordinary teacher was teaching ordinary Thuin pupils, aged 6 to 9. The building had been condemned, having been shelled during the last war. The classroom was a depressing sight: the furniture must have been at least 50 years old; there was little light; and we wondered why we had been sent there.

Then Cuisenaire took us to a table in one corner of the room where pupils were standing in front of a pile of colored sticks and doing sums which seemed to me to be unusually hard for children of that age. At this sight, all other impressions of the surroundings vanished, to be replaced by a growing excitement.

After listening to Cuisenaire asking his first and second grade pupils questions and hearing their answers immediately and with complete self assurance and accuracy, the excitement then turned into an irrepressible enthusiasm accompanied by a sense of illumination. I knew at once that here was one of those events whose significance is measured by a complete change in oneâ&#x20AC;&#x2122;s life. I knew that I had been searching in the wrong direction for years, but at the same time realized that all my work had prepared me for this momentâ&#x20AC;&#x201D;the one which brought immediate recognition of what Cuisenaire was offering to the world.

The next few hours were spent partly in arranging with Cuisenaire for his material and writings to be made available for tests in London schools, and partly in trying to formulate what I had just experienced. The teachers who accompanied me were 174

20 Discovering Cuisenaire

also most impressed by what we had seen, but were a little baffled by my inner movements, my appearance, and my language after the event.

I was considered in some quarters an expert in mathematics teaching. I had contributed as much as I could to the work of the committee of the Mathematical Association which produced the report on Primary School Mathematics. What I was now saying was that this primary school teacher, whose mathematics was insignificant and whose academic psychology simply a set of clichĂŠs, was giving us a lesson the equivalent of which could only be found ten centuries ago when the written Arabic notation was introduced in mathematics making possible the calculations of today. I realized at once how wrong I had been in accepting Peanoâ&#x20AC;&#x2122;s axioms, in believing in the elegant work of people like Landau, and in thinking that Piaget had a real knowledge of how children conceive of number. In no time my mind was unstructured allowing it to receive the news, and for years afterwards I tried to interpret the approach to learning which Cuisenaire had entered intuitively, in order that more people might understand it and so help children, as he did.

At the time I was engaged in research outside the field of mathematics, but this, too, was forced to yield before the urge created by the knowledge that I could save millions of children from the traumatisms caused by bad teaching in the early stages of schooling and replace their anxieties by joyful learning.

Moreover, I also realized that the bottlenecks caused by schools failing to enable enough people to acquire an adequate degree of competence in the skills needed by society would simply vanish if those in authority only knew what I now knew. In October 175

Part V Elementary Mathematics

1953 I circularized all Departments of Education in the U.K., telling them that I was prepared to communicate what I had learnt in Thuin. Only Dundee and Glasgow invited me to do so.

Of course, I could understand that my personality, manner, convictions, and many other factors besides would make my colleagues in authority look upon this request as improper. I was claiming that the problem of teaching mathematics to young children had been solved, and was showing a box of rods as the answer. Even people with foresight were astonished at my infatuation with these colored pieces of wood.

For several years at the London Institute of Education I had practiced ‘teaching through doing’, by taking lessons with ‘C’ streams in grammar-schools, in order to prove that method had a wider role to play in teaching than syllabuses and examination.

So I went to the schools, to the teachers directly, by-passing Training Colleges, Inspectors, and other filters in education.

Teachers agreed that there was something radically new in the lessons; but for me there was something completely renewing in each lesson: meeting children. As I had discarded the preconceived ideas blocking my mind I was obliged to reconsider, in the light of the facts I was now encountering, all that my experience had previously taught me. Indeed, had I not been prepared to reject any of the thoughts I had previously held simply because I fancied them, I should not have learnt the principal lesson of the last ten years.

Cuisenaire’s work has been an eye-opener. But an open eye can see what closed eyes obviously cannot see. What I learnt showed

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20 Discovering Cuisenaire

me that Cuisenaire, too, can be wrong, though this in no way diminishes his merits. nor his importance in my life. What I learnt and am still learning is that we shall only be able to serve as educators when we reach deeper and deeper into the mystery of the mind at work on itself. All of us can be convinced that we are doing good to others and most of us may be sure that the way we do it is right. Unfortunately, I feel like an apprentice— clumsy and heavy-handed when I meet children at work, and forget that the true attitude to them should be one of surrender, and not of dictation. This is far more important for me than all the other lessons I may have learnt and which others accept as valid for them. Cuisenaire’s gift of the rods led me to teach by non-interference making it necessary to watch and listen for the signs of truth that are made, but rarely recognized. I feel both very fortunate, and doomed, catching glimpses of what humanity will achieve when it takes the trouble, but only glimpses of it and cannot see it clearly. I am beginning to know children for what they are, and this makes me reject entirely the conclusions we find in current books, including those of Piaget; conclusions that disseminate prejudices when truth is at hand.

I have won the battle (embarked on single–handed, ten years ago) of making “Cuisenaire” a household name all over the world. But I do not know if I shall live to see even two handfuls of educators sensitized to the truth I now see so clearly. This truth is very difficult to express in words, but I know it as the one that gives hope for the future. I recognize that it will prevail eventually when enough educators live it and devote themselves to the task of spreading it everywhere. A real democracy must give all men of all ages the place that reality offers them; children, too, are entitled to their place, which in terms of 177

Part V Elementary Mathematics

spirituality is an enhanced one for they still have an innocence which most of us have lost. When dreaming of a world in which innocence is valued for what it is and for what it can do, I can only wish to be reborn into it in order to make it into a reality.

What I have learnt during the last ten years is to meet innocence and sensitivity as powers in children, and consequently to teach in such a way as to enter into agreement with them rather than maintain a conflict, as we all do at present. The price of our victory is the elimination of sensitivity and the substitution of cleverness for innocence.

Because the rods helped me to reach these conclusions, I feel bound to put them in the hand of as many teachers and parents as possible, leaving to destiny the completion of the task.

178

Appendix

If L is a length made of rods and TL the table of all the equivalent lengths, we can say that the most extensive basis for a mathematical curriculum can be provided by the set of TL’s for all L’s.

Each line in TL is one of the partitions of L. In my Mathematics with Numbers in Color books and the teachers books that go with them, I used the word ‘pattern’ to describe any one of the set of partitions of L considered at the same time as L. The complete pattern of L is identical with TL. The set { TL } contains the set of situations, made with the colored rods or any other material, to represent a set of lengths connected by equivalence, or through algebraic operations such as addition, repeated addition (or iteration), or through properties such as permutations or permutations of permutations, or any other relationship that we manage to isolate and consider per se for a moment. This way of operating we shall call the mathematization of situations made with the rods.

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The contents of the table considered here can easily be seen to fall into four headings: 1

columns R1 R2 R3 R4

2 the words in each column; 3 the arrows linking words; 4 the fusion of two columns at the bottom of the table.

1 The initial R stands for restriction. Though there are more than 4 principles of restriction, only these were considered necessary to provide the essence of the various constructions that would produce chapters of mathematics and hence sections of any elementary mathematics curriculum.

R1 appears first because of an arbitrary personal choice and not because of a logical or structural precedence over others. Its definition is in the equivalence of L and the maximal set of rods forming a train equivalent to L. It is composed of white rods. It is also the first and last line of the TL starting with L or one length and ending with the same length made of the greatest number of rods.

2 R2 is another restriction on the set of partitions. It is formed by ignoring all possible permutations of a set of rods in each line of TL.

180

Appendix

3 R3 is the restriction on TL which allows only lines formed of two rods; these lines are thus complementary to each other in L. 4 R4 is the last restriction on TL we must consider before we can study the main ideas of elementary mathematics. It is formed of all iterated lengths; that is, each line will contain as many rods of length < L as are needed to form L exactly, or most nearly form it.

Having selected these four ways of producing basic ideas, I chose the words in each column to represent the various interpretations permissible, or the various chapters that can be begun, when only the concept in the restriction is being used.

R1 clearly provides the classical foundation for arithmetic since the repetition of the unit has served as a basis for the elaboration of integers, which then form sets that are recognized to serve for cardinal and ordinal purposes. But in R1 we recognize that, if instead of considering numbers we consider the sets formed by the white rods equivalent to the successive Lâ&#x20AC;&#x2122;s, other possibilities also exist.

R1 indicates how the rods could serve to generate the traditional approach to mathematics in schools. Thus, by ignoring all the other rods and what they can bring, we can provide a curriculum equivalent to the classical treatment of number. This shows us that we cannot lose much of what traditionalists want, since this is equivalent to a very special use of the { TL }; moreover, we

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may expect a great deal more if we can introduce other ideas and other materials with new uses.

R2 indicates how the simple fact of noticing that the same rods may appear in various lines of TL can generate new insights into how sets can be related. If we agree to use any one line to represent the set of all the permutations of the rods in that line, we shall have generated many examples of correspondences 1 to 1, 1 to 2 or 2 to 1, and so on. The modern notion of function in analysis can thus be brought to the notice of beginners as a special awareness of a very immediate kind: the multiple correspondences between elements of sets. But because they are at hand, the permutations thus singled out can be studied, and likewise their transformations which lead to a study of the group of substitutions. Combinatorial algebra in its other aspects is also open and requires no new techniques of computation.

R3 is third simply for reasons of order and display; it is more primitive than R1 or R2. Here, the study of complementarity is taken as far as it seems possible at the elementary level. It could be seen that the set of R3â&#x20AC;&#x2122;s if combined with itself, generates the set of TLâ&#x20AC;&#x2122;s, thus telling us both how fundamental complementarity is, and also that indeed nothing is lost of analysis at this level if we substitute the study of { R3 } for that of { TL }.

But because R3 is based on addition, the content described is a development of aspects of this operation.

182

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R4 is, in short, a study of multiplication and what new it generates that is not directly visible in addition. In this table there are three features that need to be brought into relief: •

the place of measure;

•

a restriction within a restriction or Ŕ4.;

•

the mutual impact of R3 and R4.

Measure, in the work with the rods, is borrowed from physics and introduces counting by the back door, since it is necessary to know how many times the unit has been used to associate a number with a given length. But measure is also the source of fractions and mixed numbers, and will serve later to introduce real numbers. Thus measure is a more powerful tool than counting, which it uses as a generator of mathematics. Counting is met in R1 and can be interpreted again as being a measure with white rods. Measure is naturally also an interpretation of iteration and is the basis of R4 and all its consequences.

If { R4 } is produced out of { TL } by a restriction to the consideration of iterations, Ŕ4 still restricts the set by retaining in it the measures that are ‘successful’ or do not require a remainder. This way of creating chapters by restriction on restriction is obviously present in many points of the table, though not singled out expressly by a name.

Finally, the bottom part of the table indicates another procedure, that of relating the two processes of R3 and R4 through a special link, here shown as the distribution law or a

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new axiom. It opens up a number of more structured chapters which go to form the complex body of elementary mathematics and is therefore the aim of study in elementary schools.

To conclude this explanatory sketch of what has been put into the table, a word can be added about the arrows. They serve two purposes: they not only indicate routes from one place to another, but also link these places together, making them logically dependent. It would be an interesting exercise (and one which I shall leave to the reader) to find out how many different routes can be produced using the arrows provided in the table.

Some of them could serve as alternative syllabuses (or curricula) to use with different classes for the conquest of mathematics.

What matters here is that the reader can see how to structure a syllabus. What must be watched is the order needed for the presentation of the notions and techniques; this is provided by the arrows and the stations (or chapters) in the course of study.

Let us note that other uses of the rods exist which are not mentioned in the table, but which could be the object of additional restrictions. Thus, the study of simultaneous equations could, for instance, form an R5.

184

Appendix

185

The Sources of this Volume

“Le Moniteur des Instituteurs”, Tamines, Belgium “New Era for Home and School”, Journal of the N.E.F., London “Times Educational Supplement”, London “Il Centro”, Bulletin of the Centro Didattico Nazionale di Studi e Documentazione, Florence “Mathematica & Paedagogia”, Journal of the Belgian A.T.M., Morlanwelz, Belgium “Le Courrier de la Recherche Pédagogique”, Journal of the National Pedagogical Institute, Paris

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For the Teaching of Mathematics – Volume 3

“The Mathematics Washington, D.C.

Teacher”,

Journal

of

the

N.C.T.M.,

“The Arithmetic Teacher”, Journal of the N.C.T.M., Washington, D.C. “Impulse”, published by Mitchell Engineering and the Journal Press, London, 1957 “Le Materiel pour l’Enseignement des Mathematiques”, Delachaux et Niestlé, Neuchâtel & Paris, 1958, published by the I.C.S.I.T.M. “Education”, Journal of the Department of Education, New Zealand “The B.C. Teacher”, official organ of the British Columbia Teachers’ Federation, Canada “Mathematics Teaching”, Journal of the A.T.M., U.K. “Cuisenaire News”, published by the Cuisenaire Co. Ltd., Reading

188

Index

Action, 3, 22, 40, 58, 60, 61, 63, 67- 73, 80, 81, 84, 86, 96, 120, 122, 142, 143, 144, 162, actual, 58, 68, 80, 81, 98 mathematization of, 162, 179 virtual, 58, 60, 66-69, 74, 80-82, 84 Actual, 10, 24, 50, 54, 58, 60, 63, 66, 68, 70, 71, 79, 80, 81, 82, 85, 98, 114, 156, 161, 162, 163, 165, 168, 169 Algebra, 3, 10, 27, 33, 34, 35, 37, 38, 39, 46, 47, 48, 62, 63, 83, 84, 94, 95, 96, 97, 98, 99, 100, 109, 115, 148, 150, 154, 170, 179, 182 before arithmetic, 3, 94 Applications, 39, 76, 133 Areas, 39, 82, 115, 142

Arithmetic, 1, 2, 3, 10, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 33, 35, 37, 38, 39, 40, 43, 45, 54, 59, 60, 61, 62, 65, 69, 70, 71, 72, 73, 74, 75, 77, 79, 81, 83, 84, 85, 86, 87, 94, 98, 99, 100, 101, 102, 105, 109, 115, 117, 120, 121, 122, 125, 126, 129, 148, 149, 150, 151, 154, 169, 170, 181, 190 mental, 2, 3, 19, 22, 26, 27, 28, 33, 40, 46, 48, 52, 53, 54, 58, 62, 66, 67, 68, 69, 71, 72, 74, 76, 80, 81, 91, 100, 102, 108, 113, 115, 127, 128, 145, 155, 161, 182 structural, 153, 154, 180

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qualitative, 45, 48, 56, 60, 71, 72, 73, 74, 83, 154 Awareness, 22, 34, 36, 47, 58, 61, 63, 66, 67, 68, 69, 70, 71, 72, 80, 81, 83, 90, 91, 92, 94, 104, 155, 162, 168, 182

Entities, 18, 67, 93, 105, 109, 154 Equi-addition, 18 Equivalence, 2, 3, 27, 33, 36, 38, 46, 47, 61, 62, 63, 73, 83, 84, 95, 96, 112, 146, 148, 150, 163, 166, 179, 180 —A, 145 classes of, 1, 2, 3, 61, 62, 66, 83, 84, 132, 165, 166 family of, 38, 121 —M, 146 Equivalent expressions, 2 pairs, 11, 15, 27, 38, 39, 7, 53, 73, 84, 85, 122, 44, 145, 146 relationships, 9, 10, 13, 4, 17, 19, 22, 26, 27, 8, 32, 33, 40, 41, 44, 7, 59, 60, 61, 67, 70, 6, 81, 82, 83, 91, 92, 3, 94, 114, 117, 120, 21, 128, 136, 151, 155, 62, 165, 168, 170 Euclid, 94, 99, 128, 137, 167 Expansion, 114, 128, 146

Bases of numeration, 3, 115 Bruner, 156 Classes, 1, 2, 3, 18, 22, 49, 55, 61, 62, 66, 83, 84, 86, 103, 131, 132, 157, 165, 166, 186 Combinations, 18, 32, 35, 48, 86, 115, 138 Combinatorial calculus, 39, 149 Commutative ring, 48, 97 Contraction, 146 Decimal notation, 1 Descartes, 167 Dialogue, 27, 34, 45, 46, 48, 91, 92, 105, 151, 161 Distribution, 96, 186 Division, 2, 15, 16, 23, 36, 37, 49, 61, 74, 85, 86, 112, 127, 144, 147 Dynamics, 35, 63, 80, 90, 93, 162 Dynamism, 27, 37, 48, 68, 69, 70, 71, 136

Factor, 16, 19, 28, 32, 36, 37, 9, 52, 53, 56, 74, 85, 6, 96, 108, 127, 143, 44, 160, 162, 176 Factorization, 16, 96 Families, 19, 33, 38, 169 of ordered pairs, 38

190

Index

Field, 2, 15, 26, 44, 48, 55, 6, 58, 63, 65, 67, 70, 6, 81, 89, 94, 101, 04, 105, 129, 150, 151, 56, 165, 166, 170, 175 Fractions, 2, 3, 11, 15, 24, 27, 6, 37, 38, 51, 52, 61, 2, 74, 75, 84, 85, 86, 12, 115, 122, 126, 143, 44, 145, 146, 147, 183 Frege, 101, 105 Functional, 22, 63, 133, 155, 68 Functional syllabus, 63

Landau, 175 Lattices, 99, 136 Logic, 3, 15, 39, 41, 43, 45, 0, 55, 58, 59, 67, 68, 5, 77, 80, 89, 92, 94, 7, 99, 100, 102, 106, 17, 120, 162, 169, 180, 86 Long division, 2, 112 Mathematical beings, 68 experience, 2, 9, 11, 13, 14, 5, 18, 32, 35, 40, 41, 9, 54, 55, 57, 59, 61, 2, 65, 72, 73, 74, 75, 6, 80, 88, 93, 103, 04, 106, 107, 109, 111, 13, 115, 122, 123, 126, 28, 138, 142, 151, 155, 62, 163, 164, 165, 169, 74, 176 situation, 2, 10, 12, 16, 17, 9, 23, 28, 35, 36, 37, 5, 46, 48, 49, 56, 61, 2, 63, 69, 81, 82, 83, 4, 85, 86, 87, 90, 91, 2, 93, 95, 99, 105, 09, 111, 113, 115, 119, 28, 132, 133, 142, 148, 50, 151, 154, 156, 161, 62, 163, 168, 179 Mathematics, modern, 27, 46, 48, 85, 87, 168 structure of, 33, 37, 84, 112, 150 Measurement, 86, 96

Geo-boards, 82, 99, 129, 36, 137, 138, 139, 140, 42, 151 Group, 2, 36, 39, 44, 53, 61, 4, 85, 99, 110, 115, 25, 132, 139, 149, 182 Hilbert, 101 Images, 104, 162 Integers, 3, 36, 38, 53, 67, 4, 85, 86, 97, 105, 45, 181 Invariant, 12, 69, 99, 137 Invention of zero, 1 Isomorphic, 34, 84, 86, 95, 67, 168 systems, 63, 86 Klein, 99

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Mental objects, 68 Metric spaces, 99 Mistakes, 108, 109 Montessori, 15 Multiplication tables, 17, 37, 118, 125 Multivalent, 46, 135, 142, 144, 150

composite, 49, 73, 98 prime, 16, 36, 49, 73, 85, 87, 98, 159 rational, 3, 34, 37, 38, 76, 97, 150, 167, 169 real, 105, 167, 183 whole, 1, 3, 15, 32, 35, 38, 51, 61, 68, 75, 86, 94, 115, 122, 143, 144, 145, 166, 167

Neutral element, 48 Number, 1, 2, 3, 4, 9, 13, 14, 15, 16, 17, 18, 22, 23, 25, 26, 27, 31, 32, 34, 35, 36, 37, 38, 39, 40, 43, 44, 45, 49, 50, 51, 52, 53, 55, 57, 59, 60, 61, 63, 67, 68, 69, 72, 73, 74, 75, 76, 81, 82, 83, 84, 85, 86, 87, 94, 95, 96, 97, 98, 101, 102, 105, 109, 110, 111, 112, 113, 114, 115, 118, 120, 121, 122, 125, 126, 127, 131, 133, 136, 137, 138, 139, 143, 144, 145, 149, 150, 154, 155, 156, 157, 164, 166, 167, 168, 169, 170, 171, 173, 175, 179, 180, 181, 183, 186 â&#x20AC;&#x201D;bonds, 125, 126 cardinal, 34, 97, 126, 143, 181 complementary, 16, 126, 144, 181

One to one correspondence, 84 Operators, 3, 38, 115 Operations, 3, 15, 16, 18, 19, 23, 34, 36, 37, 38, 61, 63, 66, 72, 74, 76, 83, 84, 86, 96, 98, 109, 115, 143, 144, 145, 154, 163, 179 algebraic, 27, 33, 37, 47, 83, 84, 94, 98, 154, 179 dynamic, 14, 35, 58, 63, 0, 82, 83, 86, 90, 92, 3, 104, 113, 128, 136, 51, 162 inverse, 16, 33, 47, 48, 96 on numbers, 109 virtual, 27, 50, 54, 58, 60, 1, 63, 66, 67, 68, 69, 4, 80, 81, 86 Order, 3, 10, 11, 15, 19, 20, 7, 33, 34, 37, 38, 39, 6, 47, 50, 52, 53, 69, 0, 72, 73, 77, 83, 84,

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Index

5, 91, 93, 97, 99, 103, 04, 108, 144, 145, 147, 62, 163, 164, 168, 170, 75, 176, 178, 182, 186

0, 41, 44, 46, 47, 53, 9, 60, 61, 67, 69, 70, 6, 81, 82, 83, 91, 92, 3, 94, 96, 114, 117, 20, 121, 128, 136, 141, 46, 150, 151, 155, 157, 62, 165, 168, 169, 170, 79 algebraic, 27, 33, 37, 47, 3, 84, 94, 98, 154, 179 dynamics of, 35, 63, 80, 0, 162 of equivalence, 2, 3, 27, 3, 38, 46, 47, 61, 62, 3, 83, 84, 112, 150, 66 of order, 33, 38, 47, 182 spatial, 98, 99, 128 Rigor, 55, 94, 97, 104 Russell, 101, 105

Pairs, 3, 11, 15, 27, 38, 39, 7, 53, 73, 84, 85, 122, 44, 145, 146 of integers, 38, 53, 67, 84, 5, 97, 105, 181 ordered, 3, 27, 33, 37, 38, 9, 47, 84, 85, 144 Partitions, 3, 35, 85, 179, 80 class of, 3, 38, 47, 50, 93 Peano, 175 Perception, 40, 47, 80, 136 Permutations, 18, 73, 86, 15, 154, 179, 180, 182 Piaget, 44, 101, 156, 175, 177 Positive rationals, 84 Progressions, 18, 39, 70, 81, 15, 149, 154 arithmetical, 17, 18, 22, 8, 37, 39, 40, 60, 61, 5, 69, 70, 71, 72, 74, 1, 3, 84, 85, 98, 120, 149 geometrical, 39, 70, 82, 9, 136, 149, 151

Seguin, 15 Sensory-motor, 59 Series, geometric, 169 arithmetic, 1, 2, 3, 10, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 33, 35, 37, 38, 39, 40, 43, 45, 54, 59, 60, 61, 62, 65, 69, 70, 71, 72, 73, 74, 75, 77, 79, 81, 83, 84, 85, 86, 87, 94, 98, 99, 100, 101, 102, 105, 109, 115, 117, 120, 121, 122, 125, 126, 129, 148,

Quotient-set, 27, 47, 83, 84 Quotition, 144 Relationship, 9, 10, 11, 13, 4, 17, 19, 22, 26, 27, 8, 32, 33, 36, 38, 39,

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For the Teaching of Mathematics â&#x20AC;&#x201C; Volume 3

149, 150, 151, 154, 169, 170, 181, 190 Set, 2, 3, 10, 14, 16, 17, 21, 23, 27, 32, 33, 34, 35, 36, 39, 45, 46, 47, 48, 56, 61, 63, 66, 67, 68, 72, 74, 75, 76, 79, 81, 83, 84, 85, 86, 91, 94, 95, 96, 97, 99, 105, 109, 113, 115, 121, 126, 127, 128, 137, 138, 142, 144, 145, 150, 151, 154, 161, 166, 167, 168, 169, 170, 175, 179, 180, 181, 182, 186 algebra of, 39, 96, 109, 115, 150 idempotent, 3 Similitude, 99 Situations, 12, 17, 23, 35, 37, 45, 46, 48, 62, 63, 69, 81, 82, 83, 84, 85, 86, 91, 92, 95, 99, 105, 119, 128, 142, 151, 156, 161, 162, 163, 168, 179 arithmetical, 17, 18, 22, 28, 37, 39, 40, 60, 61, 65, 69, 70, 71, 72, 74, 81, 83, 84, 85, 98, 120, 149 geometrical, 39, 70, 82, 99, 136, 149, 151 mathematical, 3, 4, 11, 13, 15, 17, 18, 20, 26, 27,

33, 34, 40, 41, 43, 44, 45, 46, 48, 49, 55, 58, 60, 65, 67, 68, 69, 70, 74, 76, 77, 79, 80, 81, 82, 83, 90, 92, 93, 94, 97, 100, 104, 107, 116, 123, 126, 127, 128, 133, 136, 138, 142, 151, 154, 156, 162, 166, 167, 168, 175, 179 Social, 40, 56, 89, 107 experience, 2, 9, 11, 13, 14, 15, 18, 32, 35, 40, 41, 49, 54, 55, 57, 59, 61, 62, 65, 72, 73, 74, 75, 76, 80, 88, 93, 103, 104, 106, 107, 109, 111, 113, 115, 122, 123, 126, 128, 138, 142, 151, 155, 162, 163, 164, 165, 169, 174, 176 Stern, 90, 110, 154, 155 Structuration, 105 Structure, 3, 27, 33, 34, 37, 41, 46, 47, 48, 62, 70, 71, 74, 76, 81, 82, 83, 84, 86, 94, 96, 97, 98, 103, 111, 112, 123, 149, 150, 151, 154, 168, 175, 186 algebraic, 27, 33, 37, 47, 83, 84, 94, 98, 154, 179 arithmetical mental, 71 fundamental, 2, 33, 46, 48, 100, 102, 127, 182

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Index

mathematical, 3, 4, 11, 13, 15, 17, 18, 20, 26, 27, 33, 34, 40, 41, 43, 44, 45, 46, 48, 49, 55, 58, 60, 65, 67, 68, 69, 70, 74, 76, 77, 79, 80, 81, 82, 83, 90, 92, 93, 94, 97, 100, 104, 107, 116, 123, 126, 127, 128, 133, 136, 138, 142, 151, 154, 156, 162, 166, 167, 168, 175, 179 mathematical mental, 3, 74 mental, 2, 3, 19, 22, 26, 27, 28, 33, 40, 46, 48, 52, 53, 54, 58, 62, 66, 67, 68, 69, 71, 72, 74, 76, 80, 81, 91, 100, 102, 108, 113, 115, 127, 128, 145, 155, 161, 182 multivalent, 46, 135, 142, 144, 150 of algebra, 3, 34, 62, 63 of order, 33, 38, 47, 182 Sub-set, 27, 33, 47, 83, 84, 95, 97, 121 Substitutions, 39, 149, 182 Subtraction, 2, 15, 16, 17, 24, 35, 38, 47, 51, 61, 74, 85, 86, 111, 112, 122, 126, 127, 143, 144, 145, 146 with borrowing, 17

Syllabuses, 4, 10, 75, 176, 186 Symbolic thinking, 114 Transformations, 3, 39, 98, 99, 135, 182 groups of, 39, 132, 149 Unit, 10, 15, 22, 23, 38, 39, 46, 56, 59, 67, 68, 73, 74, 80, 86, 90, 102, 109, 110, 113, 115, 116, 120, 121, 127, 145, 181, 183 Univalent, 135, 136, 150 Vectors, 39 Virtual, 27, 50, 54, 58, 60, 61, 63, 66, 67, 68, 69, 74, 80, 81, 86 Volumes, 39, 40, 149 Whitehead, 101

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