Modern Mathematics: A Manual for Primary School Teachers

Modern Mathematics A Manual For Primary School Teachers

Caleb Gattegno

Educational Solutions Worldwide Inc.

First American edition published in 1963. Reprinted in 2011.

Copyright Š 1963-2011 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-266-4 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

Table of Contents

Preface ........................................................................ 1 Introduction ................................................................5 I The Algebra of the Set of Colored Rods ....................23 Sets and Sub-Sets ........................................................................... 23 Generalities .............................................................................. 23 Complementary Sets ................................................................28 Pascal’s Triangle.......................................................................30 Classes of Equivalence ............................................................. 32 Quotient-Set ............................................................................. 33 Order Relations ........................................................................34 Further Properties of the Order Relations .............................. 35 Intersection of Sets ..................................................................38 Union of Sets ............................................................................42 Properties of Intersections and Unions of Sets .......................44 Tables of Partitions ........................................................................ 47 Equivalent Lengths .................................................................. 47 Prime and Composite Numbers .............................................. 52 What Can Be Done with Ordered Pairs ................................... 54

II The Arithmetic of the Colored Rods ....................... 61 Whole Numbers ............................................................................. 61 Measuring with the White Rod ......................................................63

The Four Operations ......................................................................66 Extending the Set ...........................................................................69 Duplication and the Study of Products.......................................... 72 Division as Inverse of Multiplication............................................. 76 Large Numbers and Powers of Ten ............................................... 79 Progress on Subtraction.................................................................82 Class of Equivalences .....................................................................84 The Study of Fractions ................................................................... 87 Equivalence—A ........................................................................ 87 Fractions as Operators. Addition and Subtraction .................90 Equivalence—M ....................................................................... 93 Using the Two Equivalences ....................................................98

III The Geometry of the Colored Rods ......................105 Cubes and Prisms......................................................................... 105 Perimeters, Areas and Volumes................................................... 108 Triangles........................................................................................113

Conclusion ............................................................... 117

Preface

The first edition by Georges Cuisenaire and myself, was published in 1954 by Heinemann and Co., London. This was followed in 1955 and 1957 by two enlarged editions.

Although the successive editions were quickly exhausted, proved difficult for many readers and it seemed clear that something different was required to satisfy the needs of those who find compact writing hard to digest. Since it still appear as a separate book, this manual is intended to serve as a ‘correspondence course’ for the reader.

It is distinct from the earlier book in several ways: •

It contains none of the conceptions of the inventor of the colored rods, Georges Cuisenaire, except perhaps the ‘tables’ of each length.

1

Modern Mathematics

•

In it the material is deliberately envisaged as being ‘isomorphic’ with an algebraic system whose properties are developed in Chapter 1. In other words, this manual presents the teacher of elementary mathematics with algebra before arithmetic.

•

It is intended, not as a guide to teaching, but as a complete course for learning for the teacher.

It has taken me five years of intensive and extensive investigation in very varying circumstances, in many countries and with children of widely differing ability and background, to achieve what I believe to be a clear understanding of the intuition I had on the day I met Georges Cuisenaire, his material and his pupils. Although I then knew that the answer to all the problems of teaching arithmetic had been found, it is only today that I can present this awareness analytically. I know that Cuisenaire finds that what I am now saying is totally different from what is contained in his own method, but my debt to him is not diminished because I use only the colored rods and almost nothing of his methodology. Before I met him my own thinking led only to variations of classical themes in the field of number learning. Now I can offer a coherent, realistic and exhaustive study, free of jargon and so simple that it removes all obstacles from the teacher’s path.

All that I know has been learnt through studying the material, letting the children teach me, and answering the questions put to me by my audiences while I was propagating Cuisenaire’s

2

Preface

material. To all these people I tender my gratitude for the help I received from them.

There are still problems to be solved, but they are not the problems of teaching arithmetic.

Firstly, why is it that these colored rods are so effective?

Secondly, now that mathematics has become as natural as speech, what is to be done to meet the need for a new syllabus?

Can we design a curriculum which makes use of the children’s new power?

Thirdly, since mathematics has now found an organic place in the education of every child, can we not re-examine the rest of the curriculum with a view to broadening it and making it more functional and more human?

Tentative answers to these questions are being suggested but they cannot be included in this manual. Teachers should however be aware that because mathematics has gained its freedom, education is gaining new dimensions, and the peaceful revolution already taking place is creating opportunities for reconsidering our conception of education. Many teachers who have used the approach to mathematics advocated here have been led to reconsider their teaching in other aspects.

3

Introduction

Six years after meeting M. Cuisenaire for the first time at Thuin, I felt able to present to the teaching profession as a whole, what his inventions meant for us all.

It is often the fate of the most fruitful discoveries or inventions to give rise to a host of investigators who draw conclusions which the author had often not foreseen. I cannot call myself a disciple of M. Cuisenaire, but I am probably the visitor he impressed most and I could not resist the temptation to develop his discovery as far as my ability and knowledge allowed.

This short teachers’ manual sets out one of the aspects of my personal experience and will show the extent to which we can change a given reality according to our own experience. My contribution here (1959) is a development of the original work which Cuisenaire presented in 1952 and it includes his contribution. I hope some of my readers will in turn add other developments.

5

Modern Mathematics

I have gone around the world many times to propagate the new mathematics based, for that reason, I have worked with teachers in many different countries and accumulated an experience which I consider valuable for all of us who teach, wherever we may be.

I therefore take the opportunity of giving in this Introduction the overall picture as I know it. *

*

*

First of all, I had the opportunity to observe that, throughout the whole world, mathematics teaching seems to have been organized according to the same principles. This can be explained by the history of teaching and its expansion from the XVIth century, first in Europe, then in America, Asia and Africa with colonization. It is possible to state that its main characteristic is the development of arithmetical ideas from counting as the starting point (which leads to addition of whole numbers) to subtraction without borrowing (such as looking for the complementary numbers), to the study of multiplication tables (which contain numbers obtained by repeated additions), to the use of products in order to extend multiplication and introduce division ; finally, to the study of fractions (starting or not with decimals) and to the solution of problems.

As the whole is built up on counting, books written the world over are very much alike and only their illustrations vary according to the cultures by the introduction of some local color.

6

Introduction

The second thing any traveler can notice is that everywhere children have mathematical aptitudes quite unsuspected even by those who meet them daily. Whether in Central Africa or Japan, Australia or Scotland, Argentina or India, the young pupils come to school fully capable of entering into the mystery of knowledge; but after a few years, they generally hate mathematics or have but a very false idea of its value, so much so that, in this at least, adults seem to be impoverished children.

It has been proved over and over again that, when I teach mathematics with the rods, children quickly make incredible progress. No one can deny this, even though no-one can explain it. We usually see in our successes in school proof of the rightness of our teaching methods and in our failures the evidence of some intrinsic lack in the pupils’ mind. They are the ones who pay for our ignorance of the real obstacle and not only do they no longer enjoy studying and suffer from school and home pressure, but they become incompetent in many social functions which require that knowledge to which they have no access. They are therefore hampered, often for their whole life, by lack of numerical skill.

In some countries, educational and psychological research seems oriented towards the justification of views resulting from tradition rather than towards facts. Since such a large number of children who are classified as slow, backward or retarded prove that they can not only learn arithmetic as well as others, but often even better than the best pupils from other forms, educators feel bound to stop and ask themselves a few questions.

7

Modern Mathematics

In certain schools and school districts, teachers who have allowed the children to lead them have quickly learnt to respect the paths the child takes to reach truth. They have seen that ordinary 5 to 7 year old pupils can master knowledge which we would hardly have dared present to them at 10 in our traditional schools. Their mastery is accompanied by joy and general progress in all school subjects and also by a development of their personality, while parents do not know what to think of it all. I have met such teachers in many lands but they are still far from being legion.

In some parts of the world initiative is controlled by school authorities. These vary a great deal. When the authorities are responsible both for education and finance, they have to take double precautions and at times they start an ‘experiment’ in order to make sure that what I say is true.

In a few countries in which teachers can obtain the materials without referring back to their superiors (for example with the help of generous parents or in private schools where pupils have to buy the material as they would their books) the results they show often have a contagious effect so that the whole local community is won over.

All over the world, in a large number of centers, official experiments are in progress to test whether my claims are well founded.

In Australia, since the 1st edition of this book, most states in the Commonwealth have officially adopted the Gattegno approach,

8

Introduction

after 4 or 5 years of discussion and counter-experimenting.

Similarly in New Zealand, the Department of Education has adopted the approach officially in 1962. In Canada, two provinces have followed for three years the development of teaching with the rods; no unfavorable opinion has been expressed, whereas the opposite has often been the case.1 The Canadian public has followed with great interest the already numerous classes which have demonstrated that the pupils made better and faster progress than in the control classes working along traditional lines. In the U.S.A., the growing interest is indicated by the number of courses and seminars which, last summer, involved thousands of teachers.

In Europe, research groups are working in Great Britain, France, Spain, Italy, Switzerland and Belgium. It is true that teachers in Europe have for a long time been going ahead of their official advisers who, in some cases have not begun to look at what is developing so quickly before their very eyes. Teachers tell each other what they know and there are many more converts than we could tell. Whole districts such as Edinburgh, Glasgow and most of Scotland have adopted the rods in their primary schools and official research is following the school successes.

Spain in 1955, Portugal in 1961 and Argentina since 1959 have moved swiftly towards the adoption of this way of studying mathematics, as it permits a rapid modernization of the teaching of this subject, essential to science and technology.

South Africa, the Rhodesias, Central and Eastern African states

1 cf. Ch. Billodeau’s report published in the official gazette of the Department of Public Education, Quebec Province, April 1963.

9

Modern Mathematics

have been experimenting on a scale which speaks highly of the enthusiasm of the local teachers and administrators.

Indeed, in about one hundred countries we can today see teachers communicating to colleagues their belief that at last they enjoy teaching mathematics, through the adoption in their classes of the materials.

This picture shows that we are faced with a teaching phenomenon on a world scale where lectures and courses draw more people than the organizers ever expect to see. Teachers are good judges and if they are adopting the rods it is not because they follow a fashion, but rather because they have found what they were looking for: a solution to the problem of teaching mathematics in the primary school. *

*

*

I shall now attempt to explain the reasons for the success of the rods and to see which of their uses seem the most important to me. I have discussed this with groups of teachers hundreds of times and I think it would be useful to do it again here.

The school situation is made up of at least four factors: the material, the pupils, the teachers and the curriculum. I am not going to deal with examinations, traditions, languages and their peculiarities which are not without importance but which might confuse the issue we are considering here, already complex enough.

10

Introduction

1 Our readers are acquainted with the material and it only remains to say that though the rods themselves are the essential part of the material, the wall charts and the cards are also useful in certain cases and we feel that they must be used in order to obtain lasting and better articulated results.

2 Pupils are children of various ages who acquire that name by going to school. It is important but difficult to know them and the teacher must know certain things about the learning of arithmetic and mathematics if this learning is to be efficient.

3 The teacher is an adult who has completed certain studies, considered certain educational theories, gone through a training college and acquired an educational training as well as the opportunity of seeing other teachers teach. On the whole, teachers shelve all that and teach in the way they were themselves taught because in so doing they use the longest teaching experience they have had and know that, at least in their case, it allowed them to reach the standard of knowledge they now have. Therefore it has at least one success in its favor, whereas all the other methods seem only theoretical.

4 The courses of study are subject to a multitude of influences.

The authorities consider them as a framework which must give security to the teachers, guarantee to the public that teaching is not left to the whim of every teacher, and assure it that the pupils are being given an education useful for their life in the community.

11

Modern Mathematics

Unfortunately, inspectors and teachers consider the syllabuses as the necessary framework for an efficient education, forgetting that, on the one hand, they are the result of tendencies of the moment and that, on the other hand, they depend very much on the methods of teaching. Their content and the scheduled rate of progress cannot be separated from the use one makes of them.

Brilliant pupils can do without them, others have to struggle to make the grade or are forced to give up.

We are told that the syllabus must be considered only as a guide to be used if necessary, more particularly when the teachers are ill-equipped. But they often become a rigid framework and we should meditate on the following information: in Yugoslavia where children go to school at the age of 7, number may not be mentioned at that age without breaking the official regulations.

In certain Swiss cantons, in the first year with children of 6 or 7, according to the districts, only numbers from 1 to 19 are studied.

In Belgium, children in their first year (6 to 7) must not be taught the number 21. In England, reading and writing are taught to 5 year olds, and the teachers may teach multiplication of fractions if they wish to and are able to do so.

Time given over to arithmetic varies for children of 6, between 10 minutes a day in some countries to 45 minutes in others.

Studies on the span of attention of children prove that it is dependent on interest and, if the teaching of arithmetic has to be reduced to 10 minutes because beyond that the child is tired, it would seem reasonable to change the method. Perhaps it is verbal teaching that brings about tiredness and encourages distraction, but with the rods the children we have watched can,

12

Introduction

at the age of 5 or 10, keep ‘working’ for an hour or more and not lose interest or get tired. *

*

*

We can now advance the following considerations: If we let the children ‘have a dialogue’ with the rods and if we accept that before they see certain relationships between them they will discover more primitive ones, we shall have eliminated counting as the basis of learning arithmetic.

If we ourselves see the relation between three rods which we translate into ‘red and yellow end to end give a length equal to that of the black one’ we discover that we have made use of: 1

one operation: putting rods end to end.

2 an equivalence of length recognized with the eyes or by touch. This is not just true for the triplet of red, yellow and black, but is also true for a large number of ‘triplets’ (of rods first, then of lengths made up with rods). If we see moreover that the following way of writing translates different awarenesses in the situation presented by the black (b), yellow (y), red (r) triplet, and that it can be extended to any other triplet, we have discovered a world rather than a. fact. r+y=b

y+r=b

b=r+y

b=y+r

r+?=b

y+?=b

b=y+?

b=r+?

13

Modern Mathematics

What is left of b if we cover up a length equal to r or y? This can be written: b–r=?

b–y=?

?=b–r

?=b–y

The world we discover is the algebra of the situation. It is much easier to remember than the number triplets as a whole. Instead of the fact 2 + 5 = 7 which is unique and must be remembered, we have recognized a mental dynamism and by trying it out a few times with different triplets, we give our pupils the widest possible basis we can find without having to resort to counting which is artificial, lengthy, uncertain and requires a great deal of memory. 2 + 5 requires that I add 5 times the unit, but in 5 + 2 I add the unit only twice: this hides the equivalence which is so obvious with the rods. For 2 + 5 = 7 has to be remembered for itself, whereas in the new presentation of r + y = b or y + r = b we find that many other statements can be made by introducing different ‘measuring rods’ to the same situation: •

r = 2 white, y = 5 white, b = 7 white 2 + 5 = 7.

•

r = 1 red, y = 2 1+2

•

r=

r=

=3

light green, b=2

=2 .

pink, y = 1 +1

red

.

light green, y = 1 +1

•

red, b = 3

pink, b = 1

=1

14

pink

light green

Introduction

There is a separation of the structures and a precipitation of new relationships each time that a few are made up. This is easy and is mathematical. It is psychologically under our control; we separate questions of language and notation from those of awareness of the relationships in question.

The pupils are no longer imprisoned in the framework of numerical facts which have a meaning only if the teacher gives them one by asking questions. Here we free them by putting them face to face with what commands, decides and is essentially as multivalent as thought is.

The color of the rods plays the part of an indicator which makes it possible to speak of them without ever giving the pupil the answer. He must find it out, just as we would in the situation he is faced with. He now knows when he is right and why he is right. He no longer has to ask, trembling, ‘is that right?’ He can see it, or we can show him how he can make sure.

By reducing the role of counting we have not suppressed anything essential, since we now have more elementary and more permanent criteria. If the rods are on a table they stay there even if I turn my head away to look at a fly, whereas if I do so while I am counting I forget the number reached and have to begin again. The child therefore has less difficulty in solving his problems because he uses sight rather than hearing or in other words because we replace the role of time, which requires a great deal of attention, by that of space, which is more accessible (the spatial field of perception is, moreover, wider than that of time).

15

Modern Mathematics

By giving tangible rods to the child we have not only given him a more solid support in terms of actual manipulation but a field rich in components in which he learns that different points of view produce different results. He also sees that these different results can be recorded in different but justifiable expressions, in notations equally justifiable in their differences as in their permanencies. From the beginning he moves in a mathematical symbolism and it is a real symbolism since action and the underlying perception are never totally lost. *

*

*

The principles for reorganizing the curricula as they appear to those who study the rods are the following: • We give the child the opportunity of exhausting a situation created with the help of the rods. If at the beginning it is seen with the eyes, it soon becomes symbolical. For example, a table of lengths equivalent to a certain length could be made: that only requires the child to put rods end to end and to make equal lengths. If he writes out these equivalences using the initial letters of the colors (w,r,g,p,y,d,b,t,B,o) and the sign + to indicate that the rods are touching end to end, we have a series of additions. If one of the rods is taken away from either end of each line of the table, subtractions are produced; if the child considers lines in which he tries to make some length by repeated addition of other lengths, either he finds it always impossible and obtains examples of prime numbers, or else he reaches multiplication and division.

16

Introduction

• Once multiplication has been recognized as a separate operation it is possible to replace a row of rods by a ‘cross’.

For instance, if a row of five red rods represents five twos, then the cross formed by placing a yellow rod across a red will serve as a symbol for that product, 5 × 2. If, say, a black is put on top of this we may read the ‘tower of three’ as 7 × 5 × 2. Multiplication properties are then in their proper context, but they are always just as easily perceived.

• A special form of multiplication is seen when towers are made only of one type of rod. With these, operations on indices, or exponents, can be understood whether they give squares, cubes or higher powers, or their inverses, the roots.

A study of the tables thus obtained will serve as a basis for logarithms as in the following example which has been worked out by many children of 6 or 7.

We form 2, 2 × 2, 2 × 2 × 2, 2 × 2 × 2 × 2, 2 × 2 × 2 × 2 × 2, etc. and write the height of the tower as an exponent: 21 2 2

22 2×2 4

23 2×2×2 8

24 2×2×2×2 16

25 2×2×2×2×2 32

26 2×2×2×2×2×2 64 and we ask the question 64 ÷ 16 = ? We remove 4 rods from the tower of 64, there are two left, and the answer is therefore 22 or 4.

17

Modern Mathematics

Naturally, continuing this to 210 or further, many more questions can be asked and much more pleasure is in store. •

1

How many light green ones make the orange one? o á g = ?

2 3 light green rods and a white one make an orange one: o = 3g + w, or, o á g = 3 remainder 1w.

3 How many times can I subtract a light green one from the orange one? o – g = b, b – g = p, p – g = w: 3 times a light green with 1w left over.

4 The light green one is the light green or 3

of the orange and the orange of the light green.

of

More readings can be made if one measures each rod with the help of any other (e.g. 3 Ă— + = 1 when measuring with the black rod).

Teachers who look through my books for children, published under the title of Mathematics Books I-VII, will see the ogram

18

Introduction

we suggest so that all the possibilities of the rods are met. Using a notation and a language for each different awareness, we always bring out the mental dynamics. In this way the rods are not considered an aid to make a theory acceptable but a model of what one wants to study. The sets of rods are, in fact, according to the structures which are brought out: 1

a set of objects from which we recognize •

relations of equivalence of color and length

•

a relation of order of lengths

•

sub-sets which can be enumerated.

2 a set structured by the operation of addition which allows the study of a commutative additive group and thus serves as a model for such a group.

3 a set structured by two operations, one being the above commutative group and the other, noted as multiplication, being commutative and associative, and distributive with respect to addition. This gives a model of a commutative ring.

4 If multiplication becomes a group too, the set now appears as a commutative field, a result that explains why the rods are so useful. They are in fact a model for the set of rational numbers.

The above remarks will give the reader some idea about what is today called the -Gattegno method—Cuisenaire invented the rods and saw some of their uses, whereas I have seen others and 19

Modern Mathematics

have made a psychological and mathematical study of his discovery. One day it may become anonymous and as widespread in schools as the blackboard, no-one knowing who invented it or spread its use. *

*

*

In this book I have taken the reader by the hand and have made him discover the effect of successive structures on the set of rods. I have slowly given him the mathematical experience which he can thus acquire as easily as the children. By reading the following pages carefully the reader, whoever he is, will find that there is really nothing difficult in this attitude which is called modern because it was adopted towards 1900. If a reader finds the text difficult, it will be sufficient for him to read and manipulate the rods at the same time, as suggested, to discover how fascinating it all is. At no moment has it been supposed that the reader will need to read or study any other text whatever.

By the end of Chapter I the reader will have understood the model and worked on the structures it presents. Chapter II is devoted to numerical properties which are the substance of my pupils’ books (except for applications of which nothing is said in this manual).

Chapter III will show that these rods also provide geometrical experience which must not be neglected.

The aim of this manual is to bring each teacher to the point where he can feel that gaps in his own mathematical knowledge 20

Introduction

are filled and show him that it is possible for his pupils to have no such gaps. I am convinced that it is really easy to understand.

If the reader moves carefully and does what is suggested he will get to the end quite quickly and feel a different person for it.

Having reached the end of the journey, he will see that a new era in the teaching of mathematics has begun in which everyone can have the joy of understanding the relational universe and can create adequate models for the questions studied.

That leads us to a remark on applications. *

*

*

Most authors of school text-books or of syllabi consider arithmetic an applied science and justify its presence in the curriculum by basing it on its utility.

I share this opinion without identifying myself with it. I teach mathematics because without it pupils are handicapped in many ways in social life, but also because the world of relations and numbers is a wonderful world in itself, full of poetry and of unforeseen facts and creates better, deeper, more acute and more imaginative minds.

The field of applications is that where two or more mental structures meet: there they can be merged and the result will be solutions to the practical problems envisaged. If some are missing, one has to acquire them or leave the problem aside.

Most men live in a complex universe, and if any structure is missing, on the whole, it is mathematics. We have seen how 21

Modern Mathematics

children who know mathematics can become good engineers or scientists. This today can be the case for all our children because they will possess in their minds the mathematical structures ; they will have mastered them and also the other structures which life gives them. For those who are worrying because mathematics is not seen as an everyday reality, a glance at my Books I, II, III and VI for pupils of 5 to 7 and 9 to 11 respectively will reassure them.

I would like to finish by saying that our debt to the genius of Cuisenaire can but increase from the fact that every day we find in his rods new possibilities. Personally, I owe him a great deal, and 10 years after our meeting I still go on finding these little bits of wood, to which he has given so much life, astonishing.

22

I The Algebra of the Set of Colored Rods

Sets and Sub-Sets Generalities Take your box of colored rods and put it in front of you on the table.

This set of rods could, of course, be more or less numerous. It would still be a set. So the word set applies to a collection of objects irrespective of their number. A handful of rods is as much a set as the content of several boxes of rods.

Each rod in a set is called an element of the set.

A set may have any number of elements. If we imagine a set that has no elements, we can call it an empty set, but of course that is only a manner of speaking.

23 23

Modern Mathematics

Let us take a handful of rods—it is a set—and remove one of them; we still have a set of rods. If we repeat the operation of removing one rod, then at every stage we have a new set. Finally when all the elements have been removed we have the empty set.

On our way we shall also have met the situation in which only one element was left. This too is a peculiar set, since it has only one element. We thus say that sets can have any number of elements, including one or none.

What we have said so far is independent of the size of the original set. We have been concerned solely with the property that a set has elements and that elements belong to sets.

Another very simple fact is the following: given a set we can always choose a few only of the elements which belong to it. In a set of rods, a handful of them will be a handful of rods belonging to the original set.

Take in turn different handfuls from your set of rods; each time you make a choice you have a set of rods in your hand and a set left untouched which together constitute the original set. Each of these ‘smaller’ sets is said to be contained in the initial set; they are sub-sets of it.

Of course in any given set there are many sub-sets. This you can see by just considering the various handfuls you can obtain from one set of rods.

24

I The Algebra of the Set of Colored Rods

These sub-sets can be distinguished according to whether you retain some of the elements in two successive choices or change them all. For instance, if you take a handful and then replace only some of the rods with others to make a second handful, then the two handfuls are said to be intersecting sets. The rods retained are contained in each set and constitute their intersection.

But the two handfuls, or sub-sets, need have no element in common: they are then said to be disjoint. There is no intersection—or better their intersection is the empty set.

One handful of rods taken from the box forms a sub-set and those remaining form another. They are disjoint but together they form the initial set. They are called complementary, in the sense that if we join all the elements of these two disjoint subsets the initial set is restored without any element being omitted or repeated.

Take two handfuls of rods from your box. They are sub-sets of the initial set; they are disjoint and together they form the complement of the remaining sub-set—those left in the box. If the latter is empty the two handfuls must be complementary, each being the complement of the other.

One can of course take more than two handfuls and find that in a given set there are many sub-sets, not all disjoint and not all complementary to each other.

25

Modern Mathematics

Let us take a set made of three different rods1: say the red, the black and the blue. We can see that if we take one rod to form a sub-set we have three distinct sub-sets: the sub-set formed of the red rod only, the one formed of the black and the one formed of the blue. They are disjoint.

We can also choose sub-sets formed of two rods: the red and the black, the red and the blue, or the blue and the black. There are three distinct sub-sets of this kind but they are not disjoint two by two.

The intersection of the first two sub-sets, for example, is the subset formed of the red rod.

Each sub-set of 2 elements has a complement which is the subset formed by the remaining rod.

Thus if we have three elements we can form six sub-sets, three with one element each, and three with two. These are called proper sub-sets. The empty sub-set and the set itself, often referred to as the universal set, are called improper sub-sets, so if we count all the sub-sets (including the improper ones) we find that with a set of three elements we can form eight sub-sets.

With an original set having four elements we can see that there are four sub-sets made of one element each. But for each choice

1 In this section we shall assume that two rods are always different even if they are of the same color. When we need to consider them as equivalent we shall explicitly say so.

26

I The Algebra of the Set of Colored Rods

of one rod from the four, there remains a sub-set of three rods, just as for each choice of three rods from the four there remains a sub-set of one rod. There are thus as many choices of one as of three, or of complementary sub-sets.

To find the number of 2-element sub-sets we can proceed as follows: choose one rod, there are then 3 choices of one other. So with this one we can form three different 2-element sub-sets; they are not disjoint, of course, since the first rod is common to each pair. But there are 4 choices for the first rod giving 4 × 3 or 12 pairs in all. They include however, pairs like ‘black, blue’ and ‘blue, black’, so that there are really only half as many, or six different pairs.

The total number of proper sub-sets from the initial 4-element set, therefore, is 4 + 6 + 4 and including improper sets there are 1 + 4 + 6 + 4 + 1 = 16. i.e. 1 sub-set of 0 elements, the empty set, 4 sub-sets of 1 element each, 6 sub-sets of 2 elements each, 4 sub-sets of 3 elements each, 1 sub-set of 4 elements, the universal set.

Find the number of sub-sets that are different in a set of 2 elements, 5 elements, 6 elements.

27

Modern Mathematics

Complementary Sets Since a set can be sub-divided, in many ways, into two complementary sets, we see that for each choice of so many rods there is one choice of the complementary number and conversely. Is this true if we choose the empty set? What is its complement?

For instance, with a set of 7 different elements, if we write the number of choices of 0, 1, 2, 3, . . . 7 of these seven elements as 0 C 7 , 1C 7 , 2C 7 , 3 C 7 , 4 C 7 , 5 C 7 , 6 C 7 , 7 C 7

we can write 0 C7

= 7 C7

1 C7

= 6 C7

2 C7

= 5 C7

3 C7

= 4 C7

reading 2C7 = 5C7, for example, as ‘the number of choices of two elements among seven different ones is equal to the number of choices of five elements among these seven’.

Read each of the above equalities and test them with the rods.

Repeat this with all the choices made with an initial set having 2, 3, 4, 5, 6 elements as in No. 2 above.

What would happen if you have 8 different elements, or 9 or 10?

Having written the number of sub-sets in a set of from 2 to 10 elements, you have enough material to consider the following.

28

I The Algebra of the Set of Colored Rods

If a set has a number of distinct elements, say m, and if we take a sub-set of h elements, there will remain a complementary subset with m – h elements. So, for each choice of h elements there is a corresponding choice of m – h and, conversely, for each choice of m – h elements there is a corresponding choice of h elements. Or using the notation above: h Cm

=

m–h

Cm

Test this relation on all the examples above.

So far we have only compared complementary sets and found in some cases the actual number of sub-sets. If we form a table of all the sub-sets we have found in a given set, we can perhaps discover some other relationships between sub-sets.

29

Modern Mathematics

This triangle can be continued. Each insertion is equal to the number of sub-sets whose number of elements is at the top of the column. The number of elements in the original set is on the left.

Thus: 126 = 4C9 (or 5C9) i.e. there are 126 different sub-sets each of 4 elements, in an initial set of 9 elements.

Pascal’s Triangle The numbers of sub-sets in a set can be obtained in different ways: 1 As we did earlier: by counting the number of sub-sets having 1, 2, 3, etc. elements. Thus if we have 7 elements, there are 7 different sub-sets having one element each (and also 7 having 6 elements each). Each of these 7 can be paired with 6 different rods, so that there are 42 pairs, but only 21 of them are different, since ‘black and blue’ is not different from ‘blue and black’. For each of the 21 pairs there are 5 choices of the remaining rods to form a triplet (say black, blue, yellow), so that there are 21 × 5 = 105 triplets. But these are not all distinct since the triplet ‘black, blue, yellow’ may be the outcome of joining yellow to black and blue, or blue to black and yellow, or black to = 35 different triplets yellow and blue; there are therefore (also 35 for the complementary sub-sets of 4-elements each).

We thus find the numbers in the 8th row of the triangle.

30

I The Algebra of the Set of Colored Rods

2 We can also observe that each number in the triangle is equal to the sum of the one immediately above and the one on the left of that, so that: 2 C7 = 2 C6 + 1 C6

and 4C6 = 4C5 + 3C5

or, in all cases, p

Cm

=

pC m – 1

+

p – lC m – 1

Test this formula for all the numbers in the triangle and continue the lines up to, say, the 13th.

This triangle is given different names in different countries. It is often referred to as Pascal’s triangle.

So far we have counted the sub-sets having a given number of elements. Can we now find the total number of sub-sets in a given set? This is easily done by adding horizontally the numbers in each line of the preceding triangle.

31

Modern Mathematics

So if the number of elements is m the number of sub-sets is 2m.

Test this result within the continuation of the triangle you formed above. We have thus counted the number of sub-sets in a given set and found a formula that is easily remembered.

Classes of Equivalence In the preceding sections we considered all rods to be different, as in fact they are. But if we had the two sets (yellow, black, red),

(yellow, blue, light green)

would we say that they were disjoint or intersecting? Clearly if the yellow rod of the first set is considered to be different from the yellow rod of the second set then the two sets are disjoint.

But they could also be considered as sub-sets of the set, (yellow, black, red, blue, light green) in which case the yellow in each refers to the same element.

We shall usually be concerned only with the color and the length of rods and accept that any one yellow, say, can take the place of any other of the same color and length. We say that all rods of one color are thereby equivalent.

From this point of view the two sub-sets considered above are therefore deemed to be intersecting sub-sets of the set (yellow, black, red, blue, light green). If we place eleven rods in front of us to illustrate the set and two of its sub-sets, the repetitions will 32

I The Algebra of the Set of Colored Rods

only be an aid to the memory; the yellow in each sub-set is the yellow in the initial set.

Among the particular sub-sets we can form from a set of colored rods, there are all those groups in which the elements are equivalent: the whites, the reds, the light greens and so on.

There are ten such sub-sets, each forming what we shall call a class of equivalence. The property of these classes is that any given element of the original set belongs to one and only one class of equivalence.

Quotient-Set The original set having been divided into its ten classes of equivalence, one element of each class can serve as a representative of its class. Any sub-set formed of one element of each class is clearly a special sub-set. We shall call it the quotient-set of the original set. (This word reminds us that the original set has been divided into its disjoint classes of equivalence and as in the case of a quantity being shared out—so much for each of a certain number of people, that number being the quotient—so here this sub-set is a quotient: the relationship of equivalence being, for each class, the analogue of what fixes the amount.)

This quotient-set is a very special sub-set and we shall find many useful properties of the original set when we study it in the next chapter. Here we shall deal with the families of colors chosen by Cuisenaire in order to induce other properties in the quotient-set and through it, in the whole set.

33

Modern Mathematics

Cuisenaire makes use of the three primary colors: red, blue and yellow ; white is the synthesis of all colors, and black the absence of color. There are five color families in the quotient-set, forming sub-sets of related rods (related by their pigments): the ‘reds’ are three:—red, pink, tan, the ‘blues’ are three:—light green, dark green, blue, the ‘yellows’ are two:—yellow and orange, the black one, and the white one.

We shall see that these additional relationships increase the uses we can make of the material for specialized purposes.

Order Relations Rods of the same color are equivalent and those of different colors are not equivalent. But, instead of the attribute of color, we could use that of length, since rods of the same length have the same color and conversely, unequal rods have different colors.

There are also ten classes of equivalence if we use the criterion of length, and we have now found a new relationship between the rods of the whole set: that of order. We shall say that a set is ordered if by taking any two rods we can decide whether they are equal, or whether one is smaller than the other. If one is smaller than another, the other is bigger. As far as order is concerned it is the same whether we use ‘smaller’ or ‘bigger’ ; they express the reversal of a reading of the relationship.

34

I The Algebra of the Set of Colored Rods

Take any handful of rods, and order them, using the relationships ‘smaller than’ or ‘equal to’. Do this again with other handfuls.

It is clear that we can order the whole set we have in front of us, or any sub-set of it, or any number of sets.

Each sub-set has a first element and a last element.

It is clear that if a and b are any two rods we have only one of the three possibilities for any pair of rods: either a < b,

a is smaller than b,

or a = b,

a is equal to b,

or a > b,

a is bigger than b.

This is usually written a b which is read as: a is either smaller, equal to, or bigger than b.

A set in which this is the case is said to be ordered, but if the equality is excluded it is said to be strictly ordered.

See that the quotient-set is strictly ordered.

Further Properties of the Order Relations If we accept the axiom that two contradictory statements cannot be true, then the statement:

35

Modern Mathematics

a < or = b and

a > or = b

implies that a=b is the relationship between a and b.

If we told you that: a < b and b < c being a third element, then you can conclude that a < c.

This property of the order relation is called transitivity because it enables us to pass from two relationships to a third.

Is it true that if a < b and b = c then a < c?

Is the equivalence relation between rods transitive in terms either of color or of length?

Since equivalence is a special case of order, there are properties of equivalence which do not belong to order.

36

I The Algebra of the Set of Colored Rods

For instance a is equivalent to itself, but is neither smaller nor bigger than itself. We say that equivalence is reflexive but order is not.

If a is equivalent to b, then b is equivalent to a, but it is not true that if a < b then b < a. We say that equivalence is symmetrical but order is not.

Using color as the criterion of equivalence of the rods, can you express these three properties of equivalence? Can you see that these properties were implicitly used when we formed our classes of equivalence?

We can see that if we order the quotient-set, the white is smaller than the red, the red is smaller than the light green, etc. up to the orange. In doing this, we use one pair of rods each time, the consecutive rods. But we could take any pair and conclude which is smaller. This is obvious to our eyes.

For a more sophisticated mind we could prove this because we can decide which of two consecutive rods is smaller, and because of the transitivity of the relationship, we can always decide which of two given ones is the smaller.

We can put it like this: â&#x20AC;&#x2DC;Is the red smaller than the black?â&#x20AC;&#x2122; Since the red is smaller than the green and the green than the pink, the red is smaller than the pink. But this is smaller than the yellow, so the red is smaller than the yellow and since this is smaller than the dark green the red is smaller than the dark

37 37

Modern Mathematics

green. Finally, the red is smaller than the dark green which is smaller than the black, hence the red is smaller than the black.1 Intersection of Sets We already know that one set can contain another and that this one is then contained in the first. If we indicate the former relationship by , the latter by and write S1 S2 for â&#x20AC;&#x2DC;S1 contains S2â&#x20AC;&#x2122;, we can easily compare inclusion, the property of containing or being contained, with order of lengths. We may say that the larger includes the smaller or the smaller is contained in the larger.

For instance, take a handful of rods and call the set S1. Pick out some of the rods in S1 and call this set S2. If b is a rod belonging to S1 does it always belong to S2? Is S1 we say that S1 = S2,

S2

or S2

S1 ?

When can

1 using their elements and the relationship of belonging ;

1 An experiment would have shown at once that the red is smaller than the black, but what we have done in this paragraph is to give proof by reasoning, i.e. by using mental relationships which are assumed to be true, it follows from some of them that others are true. This pattern of work is part of mathematics. As it has been seen in such a simple situation it may not strike the reader as being important. But we shall meet it again and again and it will become familiar. Children of a certain age may find that, in order to arrange the quotient-set in a staircase, they need to pass through the stage of comparison of consecutive rods (as Piaget found in his experiments in Geneva). Transitivity of order is not an obvious experience, and a study of what we do when we order a set of rods may enlighten teachers as to the difficulties involved in this kind of reasoning.

38

I The Algebra of the Set of Colored Rods

2 comparing the sets S1 and S2 so that S1 both?

S2

or S2

S1 ,

or

Is the inclusion relation symmetrical?

Is it reflexive?

Is it transitive?

Write down what each would mean using the notation or and then, using your rods, compare your predicted result with that of your manipulations.

When one set is contained in another we can say that it is smaller than the other (or that the other is bigger).

If two sets of rods S1 and S2 are given, it may happen 1 that they have some rods in common (we say they have a meet or a common part or an intersection), 2 that they have all their rods in common (we say they are equal), 3 that they have no rod in common (we say they are disjoint, that their intersection is empty).

Let us suppose we have two sets with a few equivalent elements, a set S1 containing two yellow and three black rods and some 39

Modern Mathematics

others, and a set S2 also containing two yellow and three black but having others different from those of S1 Obviously, (since we have chosen the rods in this way), the intersection of the two sets is a set formed of two yellow and three black rods. It can be written thus: S1 ∩ S2 = {2 yellow, 3 black) and read, ‘S1 intersection S2 is equal to 2 yellow and 3 black rods’.

Can you decide whether S1 ∩ S2 is equal to S2 ∩ S1?

If we have two sets with no elements in common we may write: S1 ∩ S 2 = ø and read, ‘the intersection of S1 and S2 is the empty set’.

(The words ‘null’ or ‘zero’ set are sometimes used).

But what happens if, say, S1 S2 meaning that S1 is totally contained in S2? How can you write this fact using the sign ∩?

Can you say what is expressed by the following?

S1 ∩ S 2 = S 2 We do not, of course, need to stop at two sets. Take three handfuls of rods at random and write down what you see, using 40

I The Algebra of the Set of Colored Rods

the notations of inclusion or , and of intersection, ∩. Call the three sets you have chosen S1 S2, S3 and see whether S1

S2 ,

or S1 ∩ S2 = ø,

S2

S1 or S1 ∩ S2 = . .

S1

S3 ,

or S1 ∩ S3 = ø,

S3

S1 or S1 ∩ S3 = . .

S2

S3 ,

or S1 ∩ S3 = ø,

S3

S2, or S2 ∩ S3 = . .

From the second or fourth conclusions what can you say of S1 ∩ S2 ∩ S3, or the intersection of the three sets?

Can you say what the following express: S1 ∩ S 2 ∩ S 3 = ø S1 ∩ S2 ∩ S3 = S1, S1 ∩S2 ∩ S3 = S2, S1 ∩ S2 ∩ S3 = S3, S1 ∩ S 2 ∩ S 3 = S 1 ∩ S 2 ?

If we took any number of sets of rods we could still consider whether they are disjoint.

Start with S1 = one white rod S2 = one white and one red S3 = one white, one red and one light green up to S10 = one white, one red, . . . one orange.

Write down, using , , ∩ ,as many relationships as you can between S1; S2, S3, . . . S10.

41

Modern Mathematics

What, for instance, is S1∩ S2 ∩. . . ∩ S9 ∩ S10?

What are S3 ∩ S5 ∩ S7? S2 ∩ S4 ∩ S6? S5 ∩ S8 ∩ S10?

Can you find any two among these ten that are disjoint?

Union of Sets Take two handfuls at random and find another set such that it contains every element of each of them. In this case you do not need to take more than once each element that belongs to both handfuls.

For instance, if you have taken S1 = {1 white, 3 red, 7 yellow, 4 black, 1 blue} S2 = {2 red. 4 pink, 8 yellow, 5 black, 1 orange} then the set, S3 = {1 white, 3 red, 4 pink, 8 yellow, 5 black, 1 blue, 1 orange} obviously contains every element of Sr and every element of S2.

So S3 is a special set related to S1 and S2 in a particular way. We shall call this relation the union of S1 and S2. We may write S3 = S1 ∪ S2 and read ‘S3 equals the union of S1 and S2’.

Can you say whether S1 ∪ S2 = S2 ∪ S1?

Do we need to limit ourselves to two sets to begin with?

42

I The Algebra of the Set of Colored Rods

Take three handfuls S1, S2, S3 at random and form S1 ∪ S 2 ∪ S 3 .

Repeat this formation with different handfuls until you know exactly how to obtain the union of any number of sets.

If two sets S1 and S2 are disjoint, or S1 ∩ S2 = ø, what can you say of S1 ∪ S2?

If S1

S2 what can you say of S1 ∪ S2?

Can you say what the following express •

S1 ∪ S 2 = S 1 ?

•

S1 ∪ S2 ∪ S3 = S2? What then are S1 ∩ S2 and S3 ∩ S2 and S1 ∩ S3 ?

•

If S1 ∩ S2 ∩ S3 = ø, and S1

S2, what is S2 ∩ S3?

You now see that you can relate your handfuls to each other by looking at the elements which belong to some or all, and that the or ), of union, and of relationships of inclusion ( intersection create problems which can be translated into actual examples with the rods.

43

Modern Mathematics

Properties of Intersections and Unions of Sets Let us now try to see whether, given three handfuls, it is possible to find sets which are at the same time intersections and unions of some of the three.

Thus, what does (S1∪S2) ∩ S3 mean; and (S1 ∩ S2) ∪S3, where the brackets mean that you first operate on the sets enclosed?

Work these with actual examples and see whether there are any relations between these writings. If you find one, try other examples in order to know whether this was merely a matter of chance.

1

Form S ∩ S and S ∪ S (for the same set).

2 Form S1 ∩ (S2 ∩ S3) and (S1 ∩ S2) ∩ S3.

What do you observe? Is this true for all sets S1, S2, S3?

3 Form S1 ∪ (S2 ∪ S3) and (S1 ∪ S2) ∪ S3.

What do you observe?

Is this true for all sets S1, S2, S3?

The equalities you have just discovered in (b) and (c) are called the associative property of ∩ and ∪.

4 Form S1 ∩ (S2 ∪ S3). This relates the two operations ∩ and ∪ and we already know that the 44

I The Algebra of the Set of Colored Rods

set it represents is in general different from (S1 ∩ S2 ) ∪ S 3 .

See whether these two are equal: S1. ∪ (S2 ∩ S3) and (S1 ∪ S2) ∩ (S1 ∪ S3).

Similarly see whether the following two are equal: S1 ∩ (S2 ∪ S3) and (S1 ∩ S2) ∪ (S1 ∩ S3).

The equality is expressed by saying that (d) gives the distribution of ∩ with respect to ∪ and that of ∪ with respect to ∩.

Since you obtained all these results by comparing sets of rods, you may think that these relationships belong only to such sets.

Go through all the relationships in this section and imagine that, instead of rods, you have colored beads or objects that can be distinguished by shape rather than by color or length. Do you think the results you found still hold?

In fact, the properties we have used were that elements belonged to sets, that sets were either contained or not contained in each other, and the operations were those of forming sets having particular elements taken from other sets.

This means that all the results established with our sets of rods and their sub-sets are true of any sets, and that the rods do not necessarily play any role. In fact they do not.

45

Modern Mathematics

Look at the following relationships: X

or = Y, X â&#x2C6;Š Y = X and X â&#x2C6;Ş Y = Y

Can you see that they express the same fact? If you cannot, take your rods ; make sure that you understand the mutual equivalence of the three statements by using the rods; then think of beads or whatever objects you wish. If it still seems true to you, you have a mathematical grasp of the situation.

With sets of rods we have exemplified parts of what is called the algebra of sets. So it was obvious that many problems of a nonnumerical character can be put by using a set such as the one we have been working with. But it may be that it contains other properties, because of one property of the rods we have not yet discussed, i.e. the fact that the difference between any two consecutive rods is always equal to the white. This we shall use in the next section.

Here, let us add only that the relationships used so far may belong to all sorts of collections since they are more general than the ones that follow, (of course order does not apply to objects distinguishable only by shape). Our purpose however has been to show one more use of the colored rods and to give some insight into non-numerical mathematics by using an aid invented to help people to do arithmetic.

Historically, men first discovered numerical properties and only later did they realize that operations could be thought of as things of the mind in their own right.

46

I The Algebra of the Set of Colored Rods

Tables of Partitions Equivalent Lengths So far we have considered the rods as identifiable objects. We shall now look at them more closely and in order to do so let us remake a staircase out of the quotient-set. We can see at once that a white rod will bring any rod level with the next.

This property is a special case of another one (which in its turn results from the fact that the rods are cut in this particular way, i.e. as multiples of the white one): if we put certain rods end to end we form a length equivalent to the length of one or more rods (these also end to end).

Putting rods end to end in order to form an imaginary rod of that length is a new operation in this book. Its character will be disclosed if we look at the properties of our arrangements of rods end to end.

1 Take two rods a and b and put them end to end. If we indicate this by writing a + b, then b + a will mean that the same rods are put end to end, but that the one which was on the right is this time placed on the left.

We find that: a + b = b + a In this discovery, have we made use of the property mentioned above? If we replace a or b with another rod equivalent to it, the relationship is not altered. Thus the operation of placing end to 47 47

Modern Mathematics

end (or addition) is invariant (does not change) with respect to the substitution of one term by an equivalent one.

2 Now, as well as a and b take another rod c. If we write a + b + c for the three rods end to end, from left to right, is it true that b + a + c = a + b + c?

Can you make all the possible arrangements of these three rods end to end which give the same length (the same imaginary rod)?

We can say that â&#x20AC;˘

addition has the property of commutativity (i.e. is not altered when the order of the terms is altered),

â&#x20AC;˘

if we add an equal length to each of two equal lengths the resulting lengths are still equal.

Instead of only comparing two or more rods with themselves (which seems to suggest equality in advance), if we take our rods at random and form as many different arrangements as we can of different or equal ones, but so that all lengths obtained by putting rods end to end are equal, we form a table of that length.

For instance, the table of the red rod is formed of a red rod, and of two white ones, and of nothing else.

48

I The Algebra of the Set of Colored Rods

The table of the light green rod is formed of a light green rod, a red and a white, a white and a red, three white ones and of nothing else.

Make tables of each rod up to, say, the dark green.

In these tables, the choices of the rods are dependent upon the way they are cut and hence now, for the first time, we are making use of the property mentioned at the beginning of this section.

If you start with any length and try to find all the different ways of making it by using other smaller rods, you obtain what is called the table of partitions of that length. Let us try to find all the partitions of the red, the light green, the pink, the yellow, the dark green, the black, leaving out rearrangements of a given group of colors (each such arrangement is called a permutation if all the rods are different.

We can learn how to calculate the number of these permutations so as to find all the partitions possible when the order of the rods is also taken into account).

49

Modern Mathematics

It is clear that as the length increases, the table grows rapidly, and we shall need a very large number of rods to complete the table of the partitions of the orange rod.

We now want to try to discover all that can be said about our tables independently of the length with which we start. These properties of the sets of equivalent lengths, formed of rods end to end, are important for they cover a large number of examples and will teach us many things at once, thus extending our reach much more widely than when we find one property for this length and another for that. The method we are using here is algebra because it is concerned with how we do rather than with the special case we have in front of us.

It is possible to produce each line of the table either by following particular procedures or by taking the rods we put end to end at random.

Start with any length and make as many lines as you like, placing them contiguously so that you get a rectangle one side of which is the starting length. The table does not need to be complete before you make some interesting observations.

You may find that just below the rod at the left or right end of one line there are others which end to end form a length equal to that of this rod. This can be interpreted in two ways 1 in each partition (or line) a group of rods end to end can be replaced by their sum,

50

I The Algebra of the Set of Colored Rods

2 from any partition we can get new ones by replacing one rod by groups of rods whose sum is equal to it.

This of course applies to each of the rods forming one line, except the white rod.

In particular, if each rod is replaced by a group of white rods we obtain a special line, all white, which gives the cardinal number corresponding to that length.

Let us note at once the difference between the line giving the cardinal number of the length and the first line, which provides the length as a unit, i.e. the length measured by itself.

Between this line and the white one, any partition contains rods forming sub-sets of the set exemplified by the given length, and the relation of inclusion will be in terms of comparison of lengthsâ&#x20AC;&#x2122;, if a rod or group of rods end to end is longer than another (which is judged by the senses), the first is said to contain the other. Thus, since the red and the black end to end are equivalent to the blue, we can say that the length of the blue contains that of the black and contains that of the red.

Two rods, or two lengths, which end to end form a given length, are said to be complementary with respect to that length. Thus in the blue, the black and the red are complementary.

Find pairs of complementary rods for each rod.

51

Modern Mathematics

Prime and Composite Numbers When we attempt more systematic arrangements, new facts appear.

Starting with any length, try to form it by using rods of one color only (but not the white) or by the same group of rods repeated.

Either this can be done or we need to use another length to complete the line.

We find that this other length added at the end is always smaller than the rod, or group of rods, used.

If it is impossible to make a given length by putting rods of one color (or groups of rods) end to end a number of times (this we shall call iteration), this length will be called a prime length or number.

The light green, the yellow and the black rods are prime lengths and you can try to find others whose table of partitions does not contain a line of one color (other than the white) or the iteration of the same group of rods.

The orange and the white end to end, and the orange and black give such lengths; find all those that are prime up to the length made by three orange rods end to end.

Besides the prime lengths there are those whose table of partitions contains lines of one color. These are called composite numbers or lengths.

52

I The Algebra of the Set of Colored Rods

We can of course consider a section of the table of each composite length formed only of these lines of one color or of the iteration of one group of rods.

Thus the length formed of two orange and one pink can be made by using only red, or light green, or pink, or dark green, or tan, or orange-and-red rods, end to end.

By forming such partial tables of composite numbers, we can find new properties of the composite lengths 1

If, instead of putting the rods of each line end to end, we place them side by side, we form rectangles and, exceptionally, squares.

2 The rectangles can be paired and the pairs put on top of one another, covering each other exactly.

3 When the result is a square, there is only one.

In the example given above, the light green and the tan rectangles can be paired, as can the pink and the dark green, the red and the orange-and-red.

The blue rod alone provides a light green square only. The orange and dark green provide two superimposable rectangles, one tan and one red, and a pink square.

We can call squares all the lengths whose partitions contain a line made of rods (or groups of rods) which form a square when

53 53

Modern Mathematics

placed side by side. They are also those whose restricted tables contain an odd number of lines made of iterated lengths.

It is clear that in this way we have already discovered interesting properties of partitions and obtained criteria for classifying the various lengths that can be made with the rods.

We have distinguished the prime lengths and the composite ones.

Among the composite lengths we have found the squares and the others.

Each composite length provides rectangles whose sides are the rods whose iteration forms that length. The lengths of these sides are called the factors of the initial length.

Choose any composite length and find its factors.

We already know that factors are paired, by the fact that they form rectangles which can be superimposed. In squares a factor must be paired with itself.

In the next chapter we shall draw more conclusions from these facts. Now we shall consider other general properties.

What Can Be Done with Ordered Pairs In the various sections above we have considered different ways of looking at different organizations of the set of rods.

54

I The Algebra of the Set of Colored Rods

Here we shall introduce yet another approach and see what can be learnt from it.

Let us imagine that we have two sets of rods, S1 and S2. If we take one rod from one set and then one from the other we say we form a pair.

If our sets of rods increased so that we can have as many rods as we want in them and if we form all possible ordered pairs we get a set called the Cartesian product of S1 and S2 usually denoted

by S1 × S2.

If S1 and S2 represent the same set S, then S × S is the product of S by itself. That is to say, S × S is the set formed of all possible pairs, the members of each pair being elements of S. Notice that the sign × does not convey the common meaning assigned to it in the multiplication of numbers. It stands, as stated, for the selection of a set each element of which is a pair, one member of the set S being associated by the pairing with another, or the same, member of S. If we consider two lengths end to end and their sum, we see that the pair we start with is an element of S × S and their sum an element of S. So addition can be considered as a correspondence (or what is called a mapping) of S × S upon S. If we consider one pair of lengths and another pair which can be defined as equivalent, this equivalence is a correspondence of S × S upon S × S.

55

Modern Mathematics

In this section, we shall find out more about this last sort of correspondence.

First we shall decide that when we select a pair of rods (a,b), we shall keep to an order so that (a,b) and (b,a) will be considered as different, unless a = b. This is what is called an ordered pair and we shall always start from the left.

It is clear that if a = a’ and b = b’, (a,b) and (a’,b’) cannot be distinguished. Conversely, we shall say that two pairs are equal if the terms in the same positions are equivalent. For instance (red, black)=(red, black) though the two red and the two black rods are only equivalent.

To make the length of any given rod we may take another rod of the same color, thus forming pairs (1,1) all having the same property. We shall call equivalent (not necessarily equal) two such pairs in which the two elements are interchangeable without the pair being altered, and we shall write (a,a) = (b,b) = . . . the equality sign referring to the property that ‘belongs’ to all pairs.

Equality and equivalence are therefore two different concepts, the first covering the identity of the elements compared, the second only referring to the fact that some property is shared by two pairs, here that the first term is equal to the second in each pair.

56

I The Algebra of the Set of Colored Rods

It may happen that to make a length we need a certain number of times another length. This property of the two lengths, of being related by that number, can also be considered as a relation of equivalence.

Thus, if (a,b) is such that a = n × b or b = m × a, n or m fixed, all those pairs satisfying the relationships above will be considered to be equivalent.

If (a,b) and (a’,b’) are such that a = nb and a’= nb’, then we shall write (a,b) = (a’,b’); the sign = may be read as ‘equal’ but meaning only equivalent (reserving equal for identical). (red, dark green) = (light green, blue) since 3 red rods = 1 dark green and 3 light green = 1 blue.

We thus find by this procedure a method of producing any number of (different) pairs having the same property. This class of equivalence is one example of what can be done. By using any values for the number n or m above, we can produce as many classes of equivalence, different one from the other, as we choose.

Another way of forming classes of equivalent ordered pairs could be to put together all those such that their two terms are equal to the multiples of a given length. So instead of a = nb or b = ma we could have a = n × c and b = m × c, n and m fixed, and c being another length.

57

Modern Mathematics

Thus (red, light green) = (pink, dark green) since red = 2 white, light green = 3 white; pink = 2 red and dark green = 3 red rods.

This property creates the class for here too (red + pink, light green + dark green), obtained by adding the left terms together, and the right together, shares the property red + pink = 2 light green, light green + dark green = 3 light green.

These two properties using whole numbers define pairs that form classes of equivalence. We shall call each class a rational number and make the convention that if (a,b) is such that a = nb, then (b,a) will be represented by and be called the reciprocal of n. If (a,b) is such that a = m × c and b = w × c, then we shall write (a,b) = and (b,a) = since c varies from pair to pair in the family but m and n remain the same. is the reciprocal of and is the reciprocal of When two pairs belong to the same family we can observe the following: From (a,b), = (a’,b’), it follows that 1

(a,a’) = (b,b)

2 (b,a) = (b’,a’) 3 (a’,a) = (b’,b) The first is called exchange of mean terms in the proportion (this is the name for two equal pairs considered together); the second expresses the equality of the reciprocals of two equal

58

I The Algebra of the Set of Colored Rods

pairs, and the last is the reciprocal of the first or the exchange of mean terms in the second.

From two pairs of a class of equivalence we can construct other classes by simple means.

Thus from (a,b) = (a’,b’) we form 1

the family of (a + b, b) = (a’ + b’, b’)

2

„

(a, b + a) = (a’, b’ + a’)

3

„

(a – b, b) = (a’ – b’, b’)

4

„

(a, b – a) = (a’, b’ – a’)

5 6

„

(a + b, a – b) = (a’ + b’, a’ – b’) „

(a – b, a + b) = (a’ – b’, a’ + b’)

To make sure that all these equivalences are true it is enough to assume that (a,b) = (a’,b’) = ’ and note that each family is equal to a relation in which only m and n appear.

Thus (a – b, a + b) = (a’ – b’, a’ + b’) because each equals *

*

59

*

Modern Mathematics

For more properties of sets of ordered pairs, see Book IV, Part II of Mathematics in which the treatment is more properly of fractions.

In this section we have seen that more algebra can be found in the set of rods by using the idea of ordered pairs and defining certain equivalences. Since others than those given above can actually be defined, we have not exploited to the full the possibilities of the rods. The reader will perhaps try further uses of them for himself.

60

II The Arithmetic of the Colored Rods

Whole Numbers It is important to distinguish two kinds of experience of numbers. One is concerned with labels and perception of elements in everyday life; the other is mathematical, i.e. operational throughout.

Just as I say that this is a bottle and a glass, because I see them, I say that there are two objects in front of me, meaning that I perceive two fields of different impressions in my whole field of vision. There is an experience of number that goes with the sensations created by a set of objects. But this experience, though it has its own nature, cannot be said to be more abstract than where objects are considered. If we are shown a variety of objects, before we can say what they are we have to adjust to each of them by concentrating our attention on each in turn. If now, at the same time as we experience the various objects, we refer in our consciousness to the act of concentration, we can see 61

Modern Mathematics

how the experience of number takes place. It is in our own psychological time, a special memory that this object has been perceived. In saying that there are 8 objects, I am saying that I have had to concentrate on each in turn and that these alternating concentrations and relaxations have been recorded, and this is precisely the experience of 8. Every time such an experience is lived through we indicate it by the name 8, as we say glass or bottle when we perceive these objects.

Number in everyday life therefore is an experience and to state the number of objects on a plate is in no sense arithmetic, since it would not be arithmetic if I said: On this plate I see two nuts, one date, a banana and four grapes. But it would be arithmetic to say: If I give you half my eight sweets, each of us will have four.

The connection is therefore not with the sound of numbers or of the number of objects, but with whether there are operations or not. Bringing 9 chairs for 9 people is an action resulting from perceptions, not from operations.

A criterion for knowing when an experience is mathematical is simply to say: does the actual number in question matter? If it does, it is not a mathematical experience; if it does not, then it is.

If there are 9 people for tea, I must get 9 cups. This is a domestic experience, not a mathematical one.

62

II The Arithmetic of the Colored Rods

If by halving the number of my sweets we each get the same number, that is a mathematical experience, for it will be true for 2 or 4 or 8 sweets and will not work for 3 or 5 or 9.

The purpose of this discussion is to eradicate the belief that the sounds or signs of numbers are arithmetic, and that if we want to teach arithmetic we must provide all sorts of â&#x20AC;&#x2DC;concreteâ&#x20AC;&#x2122; lifelike experiences involving numbers. A deeper analysis of this question would take us beyond the framework of this booklet.

Measuring with the White Rod When, in using the rods, we say: take 4 black rods, that is not, here, considered to be arithmetic; it is an order comparable to: take a knife and a fork, since knowledge of each of the objects in question is all that is required in both cases.

So our pupils can be given instructions of this type before there is any thought of teaching arithmetic.

We already know from Chapter I how to make tables of partitions, and it is assumed in that action that we are able to pick up the right rods and place some end to end and side by side, always watching that we observe the length to be obtained.

Now we can read each table using the names of the colors as pointers. For instance: the orange, the blue and the white, the red and the tan, are 3 lines of the table of the orange rod. If no

63

Modern Mathematics

mistake is made, the judgment of correctness is visual (or tactile for blind people).

Let us introduce a code and write: 1 for white, 2 for red, 3 for light green, 4 for pink, 5 for yellow, 6 for dark green, 7 for black, 8 for tan, 9 for blue, 10 for orange, and, + for end to end. These figures are read as is usual: one, two and so on and are the outcome of measuring the ten rods with the white, that is the unit of measurement.

The table of the orange rod, for instance, contains the following sets of signs, replacing the perception of colors or lengths: 10, 9 + 1, 1 + 9, 2 + 8, 8 + 2, 7 + 3, 3 + 7, . . . 5 + 5, 2+3+4+1...

If we indicate equivalence by the sign = we have: 10 = 9 + 1 = 1 + 9 = . . . = 5 + 5 = 2 + 3 + 4 + 1 = . . .

The exercise of transcribing in this code all the tables of the ten rods is a purely graphic-perceptive effort, not yet mathematical.

In particular we have: 2 = 1 + 1; 3 = 1 + 1 + 1; . . . 10 = 1 + 1 + 1 + 1 + 1 . . . + 1

64

II The Arithmetic of the Colored Rods

creating the bridge between counting as recognition of discrete objects and our code.

But here this writing refers to one single line of the tables, and these contain at the same time (except for 2) other ways of making the length of the rod: 3 = 2 + 1 = 1 + 2= 1 + 1 + 1 4=3+1=1+3=2+2=1+2+1=2+1+1=1+1+ 2= 1+1+1+1 and so on. Every time the rod is replaced by a longer one, the table of the new one contains the table of the previous one plus a border, plus new partitions. Thus if on one end of the table of the yellow rod (or 5) we put a white one, we obtain part only of the table of 6, because 3 + 3, 4 + 2 are not contained in the earlier table.

It is, first of all, this wealth of experience with respect to each number that will radically change the relationship of the child to numbers. Each number will gain a personality which it lacks when numbers are seen as being made only of units. Numbers also gain in complexity because so much more can be done with a table that is so much larger and so full of all sorts of properties.

The table of 10 is pictorially so different from that of 9 and so much bigger, yet traditionally it has seemed sufficient to pass from 9 to 10 by giving only the experience of 10 = 9 + 1.

65

Modern Mathematics

The Four Operations From all these actions—making the tables and transcribing them in our notation—we can draw some mathematics. This will appear when we substitute for the tables the virtual actions of which they allow.

For instance, if instead of writing 3 = 2 + 1 = 1 + 2 = 1 + 1 + 1 we put the question: find the missing term in, 3 – =1

1 + 1 + 1=

3=1+

-2=1

-1=2

2+1=

1+ 1+ =3

3- =2

1+ -3

2+ =3

3=2+

3= +2

it is clear that we are assuming that the commutativity of addition, the symmetry of equality, and subtraction as the inverse of addition, are mental structures capable of being recognized, used safely and translated into notations.

The fact that this is done by 5-year olds in England is proof of the soundness of the theory behind our use of the rods and of introducing the notation after the algebraic experiences involved.

But this exercise is a preparation also for the recognition that what was done did not really depend upon the fact that the rods were white, red and light green. This mattered, of course, when the notation was used and for the code to operate, but not otherwise.

66

II The Arithmetic of the Colored Rods

It is enough to take the table of the pink rod and put a table of similar notational questions to show that no new mechanism is involved.

So in our Mathematics Book I, Part I, we present children with addition and subtraction, multiplication and fractions when they are still studying the first 5 rods; white, red, light green, pink and yellow, or from 1 to 5.

How is this possible?

If, in a table, a line contains more than one rod of one color, we count these rods and introduce the notation x instead of +, so that 4 = 2 × 2, 4 = 2 × 1 + 2, 5 = 2 × 2 + 1, 3 = 3 × 1 are only new readings of what is already known.

Similarly, from 2 = 2 × 1 or 4 = 2 × 2 we draw 1 =

× 2 and 2 = × 4 which is a notation for reading each equality the other way round.

Again, from 5 = 2 × 2 + 1 we can ask: how many 2s in 5? and answer 2, remainder 1.

Six-year olds experience no difficulty in converting these additions into short divisions.

This is, in fact, a general approach since if we start from a table of partitions of a given length we can always

67

Modern Mathematics

1 read each line as an addition, the answer always being the same, 2 by removing one or more rods from either end, obtain the situation of subtractions, first reading them in order to find the complementary number of a given number 7 + ? = 9 or 9 = ? + 7 or ? + 7 = 9 or 9 = 7 + ? and then rereading in the form of 9 – 7 = or what is 9 – 7?

A particular partition is one in which the length is formed of rods of one color, or by the repetition of the same group of rods.

This is what we called iteration. A length which can never be made by iteration of rods, we call prime, the others we call composite, and in these each of the lines of one color provides us with what we call factors. Thus the black rod represents the prime number 7, and the tan, which has two lines of one color (the pink and the red) the composite number 8, with the factors 4 and 2.

When a prime or composite length is formed by using rods of one color first and completed with the shortest rod possible we have a situation which leads to divisions.

Thus 3 × 3 + 1 = 10 can be read: how many 3s in 10? and three notations introduced for this: 10 ÷ 3 = 3, remainder 1

68

II The Arithmetic of the Colored Rods

or

= 3, r 1

or 3

Again 2 Ă&#x2014; 4 + 2 = 10 can be read: how many 4s in 10? and presented in the three notations: 10 á 4 = 2, remainder 2

or

= 2, r 2

or 4

Extending the Set So far we have restricted ourselves to the ten rods, and we can see that, as soon as we have a notation for longer lengths, we shall have no difficulty in extending all this to the mathematical study of larger numbers.

The extension is made in three stages: 1 First we shall learn to name lengths up to two orange rods end-to-end, and study their tables of partitions; 2 Then we shall learn to name any length, but as we are concerned with knowing as much as we can about numbers, we shall devise means of gaining information without having to

69

Modern Mathematics

form all the tables of partitions, although we know that that approach could be used; 3 Then we shall see that there are two types of properties of numbers, one called additive and one multiplicative, and that whereas additive properties require the making of lengths involving the use of many rods, the others can sometimes be exemplified with only a few.

Let us begin by finding the first extensions of the properties we met with the first ten rods. It is important to recognize the essential difference between what follows and the traditional approach.

A black rod is also 7 in our code, an orange rod 10. A black and an orange end to end will give a length that we shall call seventeen and write as 17. This is the name of that length and if we make the table of the length we can write: 17 = 10 + 7 = 7 + 10 = 9 + 8 = 8 + 9 = . . . if we use only rods up to the tens. If we have 10 + 2 + 5 we can put together (10 + 2), give it the name twelve and the notation 12, so that: 17 = 12 + 5, but this is only a new way of reading, for 12 is the name of the length and the short way of talking of the table of its partitions.

We agree to call eleven, twelve, . . . seventeen, eighteen, nineteen, twenty, the lengths and the tables of partitions of which one line is respectively 10 + 1, 10 + 2, . . . 10 + 7, 10 + 8, 10

70

II The Arithmetic of the Colored Rods

+ 9, 10 + 10, and to use the notation 11, 12, . . . 17, 18, 19, 20 for them.

Since 11 = 10 + 1 = 1 + 10 = 9 + 2 = 2 + 9 = . . . =2×5+1=3×3+2=...

20 = 10 + 10 = 2 × 9 + 2 = 2 × 8 + 4 = . . . we see that we have obtained at once the required extension.

So we can now attempt to discover, as before, all the properties of these ten new lengths, and put such questions as 1

Find all the prime numbers between 10 and 20.

2 Find all the factors of the composite numbers between 10 and 20.

3 Give the answer to questions of this type: How many 6s in 19?

How many 17s in 20?

How many 3s in 15?

This is developed in Book I, Part IV, of Mathematics and in the worksheets.

Instead of continuing with the same procedure (which would be boring), we can now use the orange rod as a unit, calling the iterations when 1, 2, 3, . . . 9, 10 are used ten, twenty, thirty, . . ninety, one hundred, and writing them 10, 20, 30, . . . 70, 80, 90 100.

71

Modern Mathematics

If we put them side by side to form a rectangle instead of end to end, we see that the width of this rectangle is equal to the length of a red rod, the width of a rectangle made of three orange rods is equal to the length of a light green, . . . and in the case of nine orange rods it is the length of the blue.

So we can represent each of the rectangles formed of orange rods by a cross made of two rods, one of them an orange. Thus a yellow-orange cross will represent 50, a tan-orange cross 80, an orange-orange cross 100.1

Duplication and the Study of Products Now that we can name numbers beyond 20, we shall use duplication as a means of relating numbers together and of filling gaps in our system of lengths.

If we start by putting two tan rods end to end, we get 16 or the length of 10 + 6. By taking two 16s or four 8s we get a length equal to three orange rods and a red, or 30 + 2, which we shall write 32. If we take two 32s, we get a length equal to six orange rods and a pink, or 60 + 4, which we shall write 64.

1 Because of the way the rods are cut, this is of course not only true of the orange rods, and in Mathematics the crosses appear from the start. This method of economising on rods is a necessity for any study of numbers of any magnitude, and it should become as familiar as making tables of partitions.

72

II The Arithmetic of the Colored Rods

Thus we can see that 8, 16, 32, 64 are related. They can all be made of tan rods, either 1, 2, 4 or 8. If we form crosses, we have a red-tan, a pink-tan, and a tan-tan for the numbers 16, 32, 64.

What we did with the tan rod we could have done with any other, and we get the table of doubles:

limiting ourselves to the ten rods and to numbers up to 100.

Doubling is one method of getting relationships, and trebling is another. By trebling, and finding the names of the corresponding lengths, we get:

73

Modern Mathematics

Doubling, trebling, etc., not only the length of one rod but of two end to end will give us in particular: 11, 22, 44, 88;

11, 33, 99;

11, 55;

11, 66

12, 24, 48, 96;

12, 36, 72;

12, 60;

12, 72

13, 65;

13, 78

13, 26, 52, 104; 13, 39, 117; etc.

Having obtained as many of these numbers as we can, we can place them in groups according to their right-hand figure and obtain: 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

24

25

26

27

28

34

35

36

44

45

32 42

33

30

38

39

40

48

49

50

etc.

There are still gaps. Looking at them carefully, we find that they represent either prime numbers greater than 20, or their double, treble, etc. Thus, 23 is prime, 46 is its double and 69 its treble, and these fill some of the gaps. We know, of course, how to write them by analogy with the others and because we can form a length equal to each of them.

74

II The Arithmetic of the Colored Rods

When completing this table of numbers, we can use a different ink for all primes between 1 and 100.

Another approach could be the Table named after Pythagoras which gives products at the intersection of row and column:

On one diagonal are the squares of 2, 3, . . . 10.

The numbers between 1 and 100 which are not on this table are not all prime, e.g. 26 or 96.

It is clear that the approach to the numbers between 1 and 100 described above is different from that shown earlier and allows us to find a different set of properties.

We have not only learnt to name numbers but also, instead of finding properties of each individual number as before, we are now trying to get bonds between certain numbers which form groups with respect to their multiplicative properties.

75

Modern Mathematics

We can also give all the squares in Pythagoras’ Table. ‘Square’ reminds us of the form of the figure made by the rods when placed side by side. 5 yellow rods or 8 tan each form a square and the names for the length they form when end to end are 25 and 64 respectively.

Division As Inverse of Multiplication Factors for these numbers can also be found by a procedure different from the usual one, since we can find out, at once, some of the lengths which enter into the formation of their length. Thus 45 is at the intersection of column 5, row 9, and of row 5, column 9 and equals 5 × 9 or 9 × 5. But 9 = 3 × 3; so 45 = 3 × 3 × 5 and its factors are 3 and 15 or 9 and 5.

We can check this if we wish, by making the table, using rods end to end.

4 × 10 + 5 = 3 × (10 + 5) = 5 × 9 = 9 × 5 = 15 × 3.

We can introduce a new symbol by putting rods crosswise at more than one level. Thus a light green cross with a yellow rod across on top will read 5 × 3 × 3 or 3 × 3 × 5. If we change the order, we can have 3 × 5 × 3 and in this case there is no other arrangement. By taking them all together we get 45 ; if we combine them two by two, we get 9 × 5 or 3 × 15, 5 × 9 or 15 × 3.

In other words, each of the rods gives one factor, but the product of any two or more of them also gives factors.

76

II The Arithmetic of the Colored Rods

Sometimes the number itself is considered as a factor, and since all lengths can be made of white rods, 1 is also a factor. So as the complete list of factors for 45 we can write: 1,

3,

5,

9,

15,

45

If the numbers equidistant from the ends are paired by multiplication, they all give the same result: 1 × 45 = 45 × 1 = 3 × 15 = 15 × 3 = 5 × 9 = 9 × 5.

Once we have experience of these multiplicative bonds (and this can be started with 12 = 2 × 2 × 3 or 16 = 2 × 2 × 2 × 2), we can reverse the multiplications and obtain divisions 45 ÷ 5 = 9

45 ÷ 9 = 5

45 ÷ 3 = 15

45 ÷ 15 = 3

and all the similar but much more complicated ones where there are several rods in the crosswise ‘tower’. Assuming that we know how to name the product of 7 × 8 × 9 × 6 = 3024, we can see that this result can be obtained by multiplying 56 by 54, or 63 by 48, or 72 by 42, or 7 by 432, or 9 by 336, or 8 by 378, or 6 by 504, so we can ask 3024 ÷56?

3024 ÷ 63? 3024÷48?

3024 ÷42?

3024÷ 432? 3024÷ 7?

3024÷ 336? 3024 ÷ 9?

3024÷ 372? 3024÷ 8?

What is 3024÷ 54?

3024÷ 72?

3024÷ 504? 3024÷ 6?

77

Modern Mathematics

This is only to show how, from one situation formed of 4 crossed rods, we can obtain a certain amount of arithmetical activity.

We now come to the question of how we learn to read large numbers and how we get the answer of a multiplication. So far we have had no difficulty because we were dealing only with relatively small numbers, and the procedure followed enabled us to avoid counting and learning the tables of multiplication. But shall we not find ourselves in difficulties when we wish to work out some operations we have not yet met?

The answer is to achieve this aim in several stages 1 Although we do not study the tables we know the products up to whatever level we have practiced, which is precisely what is required by traditional methods.

2 When we considered a length formed of other lengths all equal to each other, we stopped, for convenience, at a small number, but in practice we can easily attempt each day to observe numerical behaviors and see how multiplications organize themselves according to the numbers that are present in the situation. Thus, multiplying by 25 is often replaced by Ă&#x2014; 100 and division by 4; multiplying by 32 is often simplified if doubling is used. It is not true that the mechanical multiplication procedure is always the fastest. In many cases we can distinguish between the use of calculating machines and meeting a mathematical problem with initiative. When two large numbers are to be multiplied together, those who know what they are doing are at an advantage compared with those who

78

II The Arithmetic of the Colored Rods

only apply rules. For instance, multiplication by 5997 can be replaced by multiplication by 6000 and by 3 and subtraction of the results instead of using 3 multiplications and the addition of 4 numbers.

3 The traditional multiplication procedure is based on the distributive law of multiplication with respect to addition; thus 23 × 74 = (20 + 3) × 74 = 20 × 74 + 3 × 74 = 3 × 74 + 20 × 74.

If the numbers are larger 237 × 749 = (200 + 30 + 7) × 49 = 200 × 749 + 30 × 749 + 7 × 749 And this is precisely what we would do, but only after we have examined our numbers to see which of their other properties we can use to save work and avoid the risk of making mistakes.

Large Numbers and Powers of Ten The introduction to large numbers is through the formation of towers of crossed rods. If we use orange rods, we obtain powers of 10: 10,

10 × 10 = 102 = 100,

10 × 10 × 10 = 103 = 102 × 10 = 10 × 102 = 1,000, etc.

79

Modern Mathematics

the name being given each time. But if we put on top of the cross formed by two orange rods a red or a light green or a black rod, we get, for example, 7 × 100 or 70 × 10 or 10 × 7 × 10 or 10 × 70 or 100 × 7 after all manipulations. Since this is possible for 700 or 2000 or any other such towers, we become acquainted with some large numbers and their multiplicative properties.

Now we can look back to what we did with the various groups of numbers defined first by rods between 1 and 10, then by one orange rod crosswise with any other rod giving us the decades and the means of filling in the numbers in between, then by a cross of two orange rods with a third rod crosswise, giving us the hundreds, and so on for thousands, tens of thousands, etc.

A tower is a substitute for a long line of rods, but being a substitute, it enables us to think back to lines and additions.

This transition is needed for the establishment of a wellorganized oral and written numeration by means of which we can confidently name any number.

Visual control of the heights of towers and the color of the rods placed crosswise on the orange crosses will help considerably in the solution of the problem of notation.

If we have to find a notation for a number between 1 and 1,000, all we need to do is to form crosses and read them: 785, for instance, is made of a black-orange-orange tower, a tan-orange cross and a yellow rod near by. Conversely, since we add mentally between towers, we have 700 for the first, 80 for the second, and 5 for the yellow.

80

II The Arithmetic of the Colored Rods

700 + 80 + 5 = 700 + 85 and this we write 785.

If there were several towers: yellow-orange-orange-orangeorange, blue-orange-orange-orange, black-orange and white near by, we should have: 5 × 10 × 10 × 10 × 10+9 × 10 × 10 × 10+7 × 10 + 1.

104 is written 10,000 and 5 × 10 × 1000 = 50 × 1000 = 50,000 by name.

So 50,000 + 9,000 + 70 + 1 = 59,000 + 70 + 1 also by name. But we know through our manipulations that 59,000 = 5,900 X 10, so we get: 5,907 × 10 + 1 or 59,070 + 1 or 59071.

There is no need to go through all this, since we have already introduced the notation step by step and had no need to wait until we reached this stage before dealing with the matter. We reintroduce it here to show how our method of naming a number and manipulating situations can be used to solve problems of notation.

Long before meeting such large numbers, we shall have seen (see Mathematics Book II) that if numbers are placed underneath each other in proper columns, the addition (with ‘carrying’ when it presents itself) of figures leads to the same answer as the process of naming we have described.

Long division is also a procedure, but one in which there is a new element. No one can perform very long divisions without 81

Modern Mathematics

trial and error, and without changing the situation in the working out so as to understand what is being done and to save time at each stage of the operation.

A knowledge of products, of how numbers behave, makes each trial of a partial quotient easier, in Book II we show how the experience of doubling, trebling or number bonds, makes that procedure a mathematical one.

Progress on Subtraction Impressive progress in the theory and practice of subtraction has been brought about by the use of the rods.

It is well known that because of the current notation, it happens that subtractions require two viewpoints according to whether the figures of the number to be taken away are smaller or bigger than the corresponding figures of the other. It is also well known that there are at least three accepted methods for teaching subtraction in the second case: the so-called borrowing, equiaddition and complementary methods.

What we are now going to explain is not a fourth method to be compared with the others in terms of the percentage of failures of understanding to be found in each, but a procedure that removes the difficulties and does not even suggest that there are different methods. If a number can be subtracted from another number, the writing of the answer should present no obstacle.

82

II The Arithmetic of the Colored Rods

The fault lies in that we want to subtract units from units first, and if this is not possible we must do something.

The first point to recognize is that any given number is equal to the difference between an infinite number of pairs of whole numbers. Thus 2=3–1=4–2=5–3=...

The second point is to recognize that from any difference we can get a certain number of others by subtracting the same number from each 1

73 – 27 = 72 – 26 = 71 – 25 = 70 – 24 = . . . =47 – 1 = 46

by subtracting successively one unit from each at each step; 2 73 – 27 = 70 – 24 = 66 – 20 = 56 – 10 = 46 by subtracting first 3, then 4, then 10 and again 10; 3 73 – 27 = 53 – 7 = 50 – 4 = 46 by subtracting first 20, then 3 and then 4.

With the rods, all this is obvious and can be done stage by stage until mastery is reached.

83

Modern Mathematics

Class of Equivalences Mastery of what? Of the fact that there exists a class of equivalence formed of pairs of numbers one subtracted from the other. The answer is always a member of the class and is the pair in which one of the numbers (the second) is 0.

We therefore practice intensively and extensively the two points referred to above. Starting with two rods, we look for the rod which, end to end with the small one, gives the length of the bigger one. We then remove the one we found, and by putting end to end with the original rods the same rod or rods, we obtain two new lengths whose difference is the same as before.

We go on changing the lengths and writing down the difference.

Though this is always equal to the same rod, the writing looks different.

The understanding that the writing of two numbers does not compel us to operate on them is the turning point of the theory and the practice, for from any pair of different lengths we can actually deduce a whole series of pairs without altering our result. We then practice removing equal lengths from both given lengths until we reach one pair that is either known already or which by some further addition or subtraction will lead to a known result.

Here are some examples of this technique:

84

II The Arithmetic of the Colored Rods

1 73 – 27: subtracting 20 from each gives 53 – 7, adding 3 to each 56 – 10, subtracting 10 from each equals 46.

To subtract 20 is to remove two orange rods from each, to subtract 10 is to remove one orange rod from 56 and a black and light green from the other. What is left is read as 46.

2 101 – 97. For this we can add 3 first and subtract 100, or remove 9 orange rods (9 tens) and be left with 11 – 7 which is either known to be 4 or by the addition of 3 gives 14 – 10 = 4.

3 2004 – 789. In this we see that there are 20 hundreds in the first and only 7 in the second ; we remove these 7 and are left with 1304 – 89. By adding 11 to each we get 1315 – 100 which can at once be read as 1215.

The variety of the approach, which for us is such a positive thing, may seem to the reader to be a drawback. In fact, by trying it, he will be convinced that there are only advantages in this procedure.

1 There is no waste of time because each child makes the easiest transformations first, replacing each subtraction by an easier one (easier to him, that is) and no one else can say which are the bonds he knows best.

2 All difficulties due to the notation just vanish and the famous ‘0 difficulty’ never arises.

3 Most important of all, a subtraction is no longer just a procedure for getting an answer; it is part of a class of relationships and belongs to a group of

85

Modern Mathematics

transformations that maintain the essence but alter the form. This is mathematics and has a future.

Whenever a large subtraction is given, the child will inspect it before attempting to find the answer. This inspection will suggest an addition to one, a subtraction to another, a whole series of subtractions from the left or the right to a third person.

There will be no question of hard or easy subtractions; they are all opportunities for some mental gymnastics.

When some 7 year olds were given the following subtraction: 703,401 – 678,692 they suggested such a variety of methods that it was obvious to all the teachers present that subtracting was really fun for them.

These children had worked for one year with the rods and understood numbers as such, not as rules learnt by rote. They heard seven hundred and three thousand etc. and six hundred and seventy-eight thousand etc. and knew they could replace the subtraction by 25 thousand etc. minus six hundred and ninetytwo. What they did was: 703 – 678 either by subtracting 67 from 70 and 8 from 33, or by 700 – 675 (which they knew was 25 through their practice with multiples of 25), or by adding 2 to make 705 – 680 and as they knew that 700 – 680 is 20 they saw 25 as the answer. For all of them this work was worthwhile, since they now had a much simpler subtraction 25401 – 692, which was for the most part solved by adding 8 to both numbers and getting 25409 – 700 = 24709 because 25400 was 254 × 100 for them.

86

II The Arithmetic of the Colored Rods

To sum up this section on whole numbers we must say that the inspiration of the study has been that mastery can and must be achieved, and that the rods suggest ways of learning which we failed to see because the normal notation was so rigid and our minds were dominated by it. By allowing the material to guide us, we have gained a profound insight into mathematics learning and teaching.

Although it is most useful, the cardboard material has not been described or used here because a detailed study of it is already incorporated in the Mathematics textbooks and the Teacherâ&#x20AC;&#x2122;s Introduction.

The Study of Fractions Equivalenceâ&#x20AC;&#x201D;A The section on ordered pairs in Chapter I is the new foundation for our study of fractions.

Each rational number is a class of equivalence of ordered pairs.

One of its forms is the pair such that the two elements have no common factor. It is called the irreducible form of that number.

For example, in the class (1,2) = (2,4) = (3,6) = . . .

(1,2) is the irreducible form of the rational number called one half which is also written

87

Modern Mathematics

(2,3) is the irreducible form for the class: (2,3)=(4,6)=(6,9)= . . . and is also written Because we think in terms of classes rather than of fractions, we can now eliminate many of the unnecessary difficulties we put before our pupils.

Teachers who have carefully studied Chapter I will understand why we advocate a recasting of the study of fractions, but a few further remarks can be added here.

When we introduce a fraction by giving an example, e.g., by cutting a cake into two equal parts and calling each part one half of the whole, we have in mind that this will apply to all such actions, that belongs to the operation whether we apply it to cutting an apple or a piece of soap or a piece of material. The essence of is that the whole is made of twice each part. In our language, there is a pair of things, and one is twice the other or the large quantity is twice the smaller. Donâ&#x20AC;&#x2122;t we see that this is all the pairs that show this relationship?

We start with our family of pairs which we make equivalent by the relationship. Working on our pairs will show that they actually are. all equivalent to hence since each represents we can put them together into one class and use any one whenever we need it.; This happens very early when we operate on fractions.

88

II The Arithmetic of the Colored Rods

The deliberate introduction of classes, though it sounds new, is in fact what was implicit in all operations on fractions as usually taught.

The method we use in introducing and developing the study of fractions is based upon observations already made by the learner when working on integers but now transferred to pairs of numbers.

Thus, for the addition of fractions, we only need to notice, for example, that the pair (5,6) is equal to another pair, viz (5,6) = (3 + 2,6) and write it as (5,6) = (1,2) + (1,3) again using equivalence.

Conversely, if we are asked to add two fractions 1 If they have the same denominator (the second term), we only add the numerators (the first term), because each pair expresses the fact that we measure one rod by the other, and if the measuring rod is the same, then we can put end to end the two measured lengths and express the result in terms of that measure. Thus, (5,6) + (4,6) = (5 + 4,6) = (9,6). (9,6) or its equivalent (3,2) is the fraction equivalent to the sum of the two given fractions.

2 If the denominators are different, we replace both pairs by equivalent pairs having the same denominator and then proceed as in a.

Thus for (5,6) + (2,5) we form the classes of equivalence 89

Modern Mathematics

(5,6) = (10,12) = (15,18) = (20,24) = (25,30) = (30,36)= . . .

(2,5) = (4,10) = (6,15)= (8,20) = (10,25) = (12,30) = (14,35) = . . .

We take the first pair in both lines where the second term is the same; so (5,6) + (2,5) = (25,30) + (12,30) = (37,30) which is the answer.

Fractions As Operators. Addition and Subtraction What is meant by the addition or subtraction of fractions? A fraction is a way of obtaining from one magnitude another which is comparable to it in nature but differs in extension. So each fraction can be envisaged as an operator that acts upon magnitudes. Thus, acting upon any magnitude replaces it by two equal ones of which one is selected. acting on a magnitude will replace it by three equal ones of which two are selected.

Given a magnitude, we can imagine that we operate on it with one fraction and obtain a result ; if we had operated on it with another fraction, we should have obtained another result. These two results can be put together (in one way or another) and compared with the initial magnitude. If there one fraction that could have produced the final result from the initial magnitude ?

If there is, this operator is called the sum of the other two, and is written using the sign +.

Adding fractions is therefore not the same thing as adding numbers, for what we do is to go through a complicated process of first finding the results of operations (application of fractions 90

II The Arithmetic of the Colored Rods

to the same magnitude), then putting these results together and finding the fraction which links the final result to the initial magnitude. The addition is not of fractions, but of the results of the operations. The fact that it is written as an addition hides the difficulty. The purpose is to find one fraction that will do the same job as two.

How we do it is easier to grasp than why, and we should insist on clarifying all that is involved, as has been attempted in Mathematics, Book IV. Subtraction follows the same pattern.

It is clear that we must carefully follow a path in which each step is justified by the whole pattern. This we shall now describe.

If we work with the rods, we have two ways open to us: 1 When we use tables of partitions and look at products, we can relate them by the following process. Let us start with the product 7 × 8 which is shown by the black-tan cross. It says: in one way there are 7 eights, and in the other 8 sevens. So one tan rod is of the product, and one black of the same product. If we substitute the red for the black, the product obtained is 2 × 8, which is therefore of the previous product. Thus we can easily form , , , . . . of 7 × 8, and equally easily , , , . . . of 7 × 8. Ultimately these are lengths and we can put them end to end and get a length equal to their sum. So if we want to find + of 7 × 8 (or 56), we put end to end 2 tan and 3 black and get 16 + 21 = 37. So 37 = ( + ) of 56, and 37 is comparable to 56 and is of it, so that we can write, as explained above, + = .

91

Modern Mathematics

Having done as many of these operations as we can, we become familiar with replacing two operators by one which is their ‘sum’.

As 56 = 7 × 8, we can see that or

is equivalent to

or

is equivalent to

and

, and we could have found the result

above by a direct calculation. This direct calculation should come after it has been made clear why we want to do it.

2 Secondly the rods can help us make a generalization of what we explained above in connection with operations on fractions.

Since each fraction belongs to a family of equivalent fractions, when we are asked to add or subtract two given fractions we must remember that any other member of each family will do equally well. Thus, if (2,7) and (3,8) are to be added, it is not these particular forms that matter. As we know that each pair is an actual pair of rods or lengths, the first one in each bracket being measured by the other, we can look for equivalent pairs which possess a measuring rod common to both. This is always possible since we have used all the integers in the formation of the classes of equivalence.

The measure equal to 7 × 8 or 8 × 7 is, then, to be found in each family of equivalence, on the right in the bracket, and we thus know which equivalent pairs to use: (2,7) = (16,56),

(3,8) = (21,56)

92

II The Arithmetic of the Colored Rods

Reading each pair aloud, we hear: 16 fifty-sixths and 21 fiftysixths; this means that if we add (16 + 21), 37 fifty-sixths will produce the same effect as (2,7) and (3,8) on any magnitude. It is their sum.

Subtraction follows the same pattern.

The main difference between a and b is that in a we have actually used a magnitude to work upon and in b we have left the magnitude indeterminate. Both approaches are important, and if they are properly understood the teaching of addition and subtraction will be given its true mathematical simplicity.

Equivalenceâ&#x20AC;&#x201D;M Similarly we can shed light on the other two operations, called multiplication and division of fractions.

Each rod is cut as a multiple of the white one. This, as we have found, allows of the study of the whole numbers as pairs in which the second, the measuring rod, is the white one. But it also allows us to work out some other relationships.

Thus, of the three rods, white, red and tan, the white is

of the red, of the tan, and the red of the tan. The relationship of the white to the tan can be split by using the red. So (w,t) is (w,r) of (r,t), or the white is half the red which is a quarter of the tan, and the equivalence above can be written = of , the most important word being of.

93

Modern Mathematics

This should become perfectly clear before any other consideration is introduced. There is a direct relationship between the white and the tan expressed by the pair (1,8) or .

But we could also say that the white is half the red which is a quarter of the tan. In this case the relationship between the white and the tan is obtained with the red as intermediary, once as the measuring length and once as the length measured by the tan.

If, instead of the red, we use the pink, we can say: (w,t) is (w,p) of (p,t) or because the white is

is

of

of the pink which is

the tan.

Similarly, if we use both red and pink as intermediaries, the relation of the white to the tan will be equivalent to the following: the white is half the red which is half the pink which is half the tan. In notation is

of

of

Looking back at what we have just done, we find that the link between successive pairs is always the same; the intermediary rod serves to measure one rod and is measured by another. So it is a rod with two roles, which are determined by its position in the various pairs. Thus with any three rods, a, b, c, (a,b) of (b,c) expresses that we measure a by c but through the intermediary of b. We can always say that (a,b) of (b,c) is equivalent to (a,c).

This does not mean identical ; on the contrary, it is a new way of

94

II The Arithmetic of the Colored Rods

creating a class of equivalence for (a,c), one which for convenience only we shall call multiplicative equivalence.

Let us see what all this amounts to.

1

We can use any rod for b. Thus =

of

=

of

All that is required is that b shall appear twice, once as denominator and once as numerator. But it can have any value.

So for we have the class of equivalence ;

of

;

of

;

of

and similarly for all fractions.

2 We can insert more than one intermediary rod, so that if instead of we use of , we can write = and if instead of

we use =

of

of

of of

we get of

95

of

Modern Mathematics

Hence, for every fraction, we can find another enlarged class of equivalence, formed of any such sequences. In notation we can write: (a,b) = (a,c) of (c,d) of (d,e) of . . . of (x,b) in which c, d, . . . x appear twice in the series, as the second term in one bracket and the first term in the next.

This new multiplicative class of equivalence contains the word â&#x20AC;&#x2DC;ofâ&#x20AC;&#x2122; and has been formed by starting with a given pair.

If we were asked whether there exists a pair that is equivalent to (a,b) of (c,d) where c differs from b, we might find one and see whether (a,b) of (c,d) occurs in its class.

This can be found by using the class of equivalence defined at the beginning of the section on fractions.

Since each fraction belongs to a family of equivalent ones, (a,b) can be replaced by any equivalent one, as can (c,d). There is at least one form of each in which the second term of (a,b) is equal to the first of (c,d) ; for instance: (ca, cb) is equivalent to (a,b) (be, bd) is equivalent to (c,d) but cb = bc and thus:

96

II The Arithmetic of the Colored Rods

(a,b) of (c,d) is equivalent to (ca,cb) of (cb,bd), or using multiplicative equivalence, to (ca,bd). So there is at least one pair equivalent to (a,b) of (c,d) whatever each is.

Now let us compare (a,b) of (c,d) and the equivalent fraction (ac,bd).

We see at once that (a,b) of (c,d) is equal to (c,d) of (a,b) since both are equivalent to the same fraction (ac,bd). In writing (a,b) of (c,d) = (c,d) of (a,b) we express that the operator of is commutative.

Moreover, looking at (ac,bd) we discover a rule for finding the resulting fraction: ‘one of the fractions equivalent to a fraction of a fraction is obtained by taking for its numerator the product of the two numerators and for its denominator the product of the two denominators’.

Having made this discovery, we may allow ourselves to use the sign × instead of of and read this operation as a multiplication, when in fact it is not a repeated addition. So

We must not forget that for us each rational number is a class of equivalent fractions and that the above relationship must be paired with all possible equivalent results.

97

Modern Mathematics

That is to say that sometimes in carrying out this procedure we may encounter a fraction which is equivalent to

but is not

that particular one.

Thus as

of or of is equivalent to and does not appear obtained by simply applying the rule.

Using the Two Equivalences Now that we know how to obtain a fraction equivalent to a fraction of a fraction, and that the result is a family of equivalence, we can extend this to a fraction of a fraction of a fraction, etc.

Since two fractions can be replaced by a single one, this one with the next will lead to another contracted one, and so on, until we reach the last. This means that we have to use only a. finite number of fractions in the series.

Since the notation Ă&#x2014; has been used, we shall expect the properties of multiplication to belong to the operation of replaced by Ă&#x2014;. This is easily proved and is left to the reader. We have already proved that commutativity holds. Prove that associativity is true between fractions of fractions.

Is distributivity with respect to addition or of

=

of

+

also true?

98

of

II The Arithmetic of the Colored Rods

Let us consider a special case, e.g., (a,b) of (b,a). We know it is equivalent to (a,a) which in turn is equivalent to (b,b), (c,c). . .

(1,1).

If we ‘multiply’ (a,b) by (1,1) we always obtain (a,b). Similarly if we multiply (a,b) by (k,k) we obtain (ka,kb) which is equivalent to (a,b).

Thus, multiplying a fraction by (m,m) gives an equivalent fraction.

This can also be expressed in the following way.

If we multiply a fraction (a,b) first by (m,n) and then by (n,m) the result is (a,b). In notation (a,b) × (m,n) × (n,m) = (a,b) × [(m,n) × (n,m)] = (a,b) × (m,m) = (am,bm) = (a,b).

Let us call the fraction (q,p) the inverse of (p,q). The inverse of the inverse of (p,q) is (p,q) itself.

Let us now rewrite what we had above but in a different way: (a,b) × (m,n) × (n,m) = [(a,b) × (m,n)] × (n,m) = (a,b). . . . (1)

99

Modern Mathematics

The squared bracket represents the product of two fractions, or one fraction (am, bn). This fraction is multiplied by another (n,m), which is the inverse of (m,n). It will be remembered that from the product of two numbers we defined the quotient by saying that if A × B = C, then A = and B =

is the quotient of C by B,

is the quotient of C by A. This is how division is

defined as the inverse operation of multiplication.

Applying this here, we can define the quotient of two fractions and write it as

or (a,b) ÷ or (n,m) as for ordinary

numbers. From our formula (1) we have the following: (a,b) × (m,n) =

or = (a,b) ×(m,n)

which can be read as follows: to multiply one fraction by another is the same as to divide the first by the inverse of the second, or to divide one fraction by another is to multiply the first by the inverse of the second.

Now that the four operations have been made clear and can be introduced via the classes of equivalence, there should be no need for memorizing rules and applying them by rote.

100

II The Arithmetic of the Colored Rods

The main point is the substitution of one fraction for a pair of fractions, the one fraction performing the same function as the other two together. This involves addition or multiplication in the senses made plain above.

Of course, no one knows what multiplying by means in terms of repeated addition, but when it is seen as an operator the meaning is obvious: so of something, say 5, written as $ Ă&#x2014; 5 is not multiplication. The notation is misleading and has misled most authors of arithmetic text-books. It was only after we found the structure of the fraction equivalent to a fraction of a fraction that we allowed ourselves to use the sign Ă&#x2014; and found that the operator of has the properties assigned to multiplication as repeated additions.

Similarly, division of fractions is not repeated subtractions ; it is the inverse of multiplication. By looking at things in this way, we see the famous (or infamous) rule for reversing a fraction and multiplying as being a way of reading a special case of a fraction of a fraction of a fraction: that in which the last two are the inverse of each other.

Thus, the use of the rods has allowed us to think of fractions as ordered pairs; it has made plain the idea of class of equivalence by presenting naturally several pairs with different names: (4,8) and (2,4) and (1,2) according to the way they are looked at. It has forced us to consider fractions as operators and not as bits of something, and it has eliminated the obscurities covered by the words addition and multiplication and the leaps, often unjustified, from experiments with knives to mental operations.

101

Modern Mathematics

One more remark needs to be made. There has been no mention of â&#x20AC;&#x2DC;top-heavyâ&#x20AC;&#x2122; fractions. Since (a,b) as an ordered pair is a fraction, it is irrelevant that a is smaller or bigger than b. If we want to focus upon that distinction, we shall begin a new paragraph and shall probably find new statements, for we shall have added an extra condition: (a,b) is to be compared to (c,d).

In the families of equivalents we find two fractions with the same measuring rod, and we shall say that (a,b) is smaller than (c,d) if its left-hand term in the new form is smaller than the left-hand term of (c,d) in its new form.

In particular, (a,b) is smaller than (1,1) or (b,b) if a is smaller than b. This is equivalent to the usual definition if we add that (1,1) is to be replaced by 1.

If a is bigger than b or (a,b) is bigger than 1, a contains b and something more which can be measured in terms of b.

For instance, (7,5) = (5 + 2,5) = (5,5) + (2,5) = (1,1) + (2,5) or

in the usual notation for mixed numbers.

In the historical development of mathematics, these questions have loomed large. To-day they are special cases to be treated as such and given only the place they deserve, a smaller one.

Decimals are special fractions useful only because we use a decimal notation for writing our numbers. The study of them 102

II The Arithmetic of the Colored Rods

introduces new properties because every special field allows us to add something to what has been found in the general study.

The most important thing about decimals is that once their properties are deduced from those of fractions, they show themselves as behaving in all respects like whole numbers. This leads to greater simplicity in the formulation of the rules of operations.

In Mathematics, Book IV, this has been the line of approach.

Having deduced the properties of decimals as special fractions, we go on to see that their structure makes them like whole numbers.

In the same book, we treat percentages as a notation, which indeed is all they are, but there is a chapter to help pupils to master the notation and what it can express.

Note:â&#x20AC;&#x201D;As readers can find in my Mathematics details of the study of large numbers, of L.C.M. and other numerical properties, nothing is said about them in this chapter. Its function is to clear up difficulties usually experienced by teachers in understanding elementary mathematics.

103

III The Geometry of the Colored Rods

Cubes and Prisms The rods are rectangular prisms and it would seem that it would not take long to exhaust their properties. There are, indeed, not a great number of them.

Starting with the white rod, which is a cube, we can find that there are eight vertices, six faces, twelve sides. If the pupils are left to investigate the cube for themselves, instead of being told, the discovery of these facts may well take more than half an hour. The finding of criteria for ensuring that a given answer is the correct one is in itself educational, and the childrenâ&#x20AC;&#x2122;s procedure can be most instructive to the teacher. (See Mathematics, Book II).

105

Modern Mathematics

Once the answers are found, the same questions can be put with respect to any other of the rods which are cuboids or prisms.

The new property discovered will be that these numbers are independent of the length and color of the rods and are an invariant of the family (later on it can be proved that these numbers belong to oblique prisms when the cross-section is any quadrilateral as well as when it is a square).

We see that, according to the way we hold the rod, the names of the three dimensions of the prism change. Thus, if we agree that, when we place a rod on the table in front of us, the dimension parallel to our chests is called the width, that which goes away from us the depth, and that above the table the height, it is clear that the same rod can have several denominations for its dimensions.

Find the positions of a yellow rod corresponding to the following table Width

Depth

Height

1 1 5

5 1 1

1 5 1

If we take 2 yellow rods, we have more possibilities. Find them.

Width

Depth

1 2 1

1 5 2

106

Height 10 1 5

III The Geometry of the Colored Rods

5 10 5

2 1 1

1 1 2 etc.

Take 4 tan rods, find all the possible prisms you can make, and give their dimensions in a table like the ones above.

Cubes can be made out of rods of any one color.

Make cubes and compare the lengths of the sides, the areas of the faces, the volumes, using each rod in turn as the measuring yard-stick.

If we take a dark green cube and hold the rods together by means of rubber bands, we get a model of convenient size for investigating sections of the cubes. To do this we can use a rubber band of a different color.

Find when the sections are: squares rectangles parallelograms triangles equilateral triangles right-angled triangles

107

Modern Mathematics

isosceles triangles pentagons hexagons regular hexagons.

Instead of a cube, take a prism of, say, 12 tan rods placed in three layers of four each, hold them together and investigate the sections as above Can you have sections that are triangles or hexagons ?

What are the invariants of the sections when you pass from a cube to a rectangular prism? Which are lost?

It is usually difficult for pupils to recognize that the cube belongs to the class of prisms. This is because cubes are singled out by name and prisms are not sufficiently studied. With the rods, plenty of opportunities can be provided, and focusing on invariances rather than on peculiarities will further the geometrical education of the pupils.

Perimeters, Areas and Volumes Another line of approach already used extensively in Books II and VI of my Mathematics consists in the simultaneous study of perimeters and areas, or areas and volumes.

Thus, if we take two pairs of equal rods, say black and yellow, we can make the outline of three different rectangles according to 108

III The Geometry of the Colored Rods

whether we enclose the yellow between the blacks or conversely, or place them so that alternately one corner is yellow and one black. They have different dimensions but the same perimeter.

They have the same perimeter but enclose different areas. By calculating these, and replacing one of the pairs or both by others, pupils will meet a large number of examples through which they will learn to be careful when dealing with perimeters and areas.

In particular, if we choose, say, the pairs black and white, we can so arrange them that the space between the two black rods is equal to zero. The shock experienced by the learner when this happens can be used to help him to understand the possibility of rectangles (or parallelograms) with constant perimeter but variable area, including a zero area. This can now be done with any pair of pairs of equal rods.

Conversely, starting with a composite number of equal rods side-by-side, whose top provides us with a constant area, we can show by re-arranging them that the perimeters which enclose them can vary considerably.

Similar situations with respect to areas and volumes can be created as follows, and will compel the learner to consider their relationships.

The total area of a red rod is 10 sq. cms., that of the yellow 22 sq. cms. and their respective volumes are 2 and 5 c.c. If we put these two rods end to end, their volumes are added and we find that 7 c.cs. is the volume of the new prism, while its total area is 30 sq.

109

Modern Mathematics

cms. If we put them side by side, the volume is still 7 c.c. but the total area is 28 sq. cms.

If we repeat this with the various pairs, or triplets of rods, we find that while volumes remain constant whatever the spatial placing of the rods, their area is a function of the spatial arrangement.

If we take rods to represent the walls of tanks, we can see that the same area of walls can enclose different volumes, for with 14 black rods upright we can make the following tanks: •

2 walls made of 2 rods each, and 2 made of 5. To fill in this tank we need to pile up 7 layers of 5 red rods or 70 white cubes ;

•

2 walls made of 3 rods each, and 2 made of 4. To fill in this tank, we need 7 layers of 4 light green rods or 84 white cubes.

(Note that when you use the rods as walls they are linked by one sharp edge and not by faces.)

Do the same thing with 12 tan rods, or with 10 blue, or 24 pink, etc. They will all show that equal areas of walls do not necessarily define equal volumes.

In our calculations of the areas of the tanks, we left out the bottom and the top. Let us see what happens if we include these.

In case a, since the two missing faces will be equal, and the area of each is 10 sq. cms., the total area will be 2 × (14 + 35 + 10) or 110

III The Geometry of the Colored Rods

118 sq. cms. In case b, the total area will be 2 × (21 + 28 + 12) or 122 sq. cms.

Thus we have different areas enclosing different volumes and that seems normal. If we start the other way round, by piling up 3 × 4 × 5 white cubes, what is the total area? It would be enough to count the squares and we should find: 2 × (12 + 15 + 20) or 94 sq. cms.

With those same cubes we could form the following prisms: 6 × 2 × 5 or 3 × 2 × 10. What are their respective total areas?

Here again we count the squares: 2 × (12 + 10 + 30) or 104 sq. cms.

2 × (6 + 30 + 20) or 112 sq. cms.

Thus, for the volume of 60 c.cs. we can find at least three prisms whose total areas are different.

The prisms 1 × 30 × 2 or 1 × 3 × 20 have areas equal respectively to 2 × (2 + 30 + 60) or 184 and 2 × (3 + 20 + 60) or 166 sq. cms. but as before, the same volume, equal to 60 c.c.

Can we use the rods to show that prisms with the same area may contain different volumes?

Let us start with a prism whose dimensions are 3, 8, 10 and find its total area, equal to 2 × (24 + 30 + 80) or 268 sq. cms., and take another one whose dimensions are 4, 6, 11, whose total area 111

Modern Mathematics

is 2 × (24 + 44 + 66) or 268 sq. cms.; this example shows that it is possible to find at least one. Here the volumes contained are equal to 240 c.c. and 264 c.c. respectively.

If we doubled each dimension, we should find another example, and so on. The way of finding them consists in first finding a number which has at least 4 factors, and then solving the problem which, in the above example, can be seen as follows: 24 has factors 3 and 8 and 4 and 6.

3 + 8 = 11 and 4 + 6 = 10.

If we choose these values for the third dimension of the prism, but coupling 3 + 8 with 4 × 6, and 4 + 6 with 3 × 8, we find that 3, 8, 10 and 4, 6, 11 give the dimensions of two prisms with equal total areas but different volumes.

In general, let us choose two rectangles whose areas are equal and whose dimensions are a and b and c and d respectively. We know therefore that a × b = c × d. If now, for the third dimension of the prism, we couple c + d with a and b, and a + b with c and d, the volumes will be: a × b × (c + d) and c × d × (a + b)

112

III The Geometry of the Colored Rods

These are different since c + d and a + b are formed of pairs of factors that go together but form unequal sums.1 The areas of these prisms are respectively equal to: 2 × [a × b + a × (c+d)+bx (c+d)] = 2[ab + ac + ad + bc + bd] and 2 × [c × d + c × (a + b) + d × (a + b)] = 2[cd + ca + cb + da + db] Since ab = cd, the equality is established.

Form as many pairs of prisms with equal areas and different volumes as you can.

Triangles Another type of problem is the following: Remake the squares, using rods of one color for each. Take three of them and see whether you can make a right-angled triangle having the sides of the three squares as its sides.

1 To prove this point, let N be a number having different factors which in increasing order are: 1, a, b, . . . l, m, N. We have N = am = bl = . . . with a < b and l < m. If we had a + m = b + l, then b – a = m – l; let K be that value.

Then bl = (a + K) (m – K) or bl = am + Km – Ka – K2 or K2 = K(m – a) giving K= m – a or K = O. Since K = b – a then b = m, and b, l, m, would not be different factors as assumed. If K = O, m = l and again the factors would not be different.

113

Modern Mathematics

This you could do as follows: take any three rods end to end.

Turn the two on either side until they meet to form a triangle.

Then add as many rods as are needed on each side to form the three squares. Is, for instance, the triangle black, yellow, tan right-angled? and the tan, orange, dark green?

Looking at your various triangles with their squares, you can compare the areas of the squares either by calculating them or by trying to cover the largest one using the other two.

Try this with the following figures: 1

triangle and squares: red, yellow, pink.

Is there an acute angle? an obtuse angle? a rightangle?

2 triangle and squares: blue, tan, black.

Is there an obtuse angle? Is it true that each square is smaller than the other two together?

3 triangle and squares: pink, black, orange.

Is there an obtuse angle? Is it true that the square opposite that angle is bigger than the other two together?

4 triangle and squares: pink, yellow, light green.

What are the angles? Compare the area of the largest square and those of the other two.

114

III The Geometry of the Colored Rods

Can you state a relationship between squares and the angles that are opposite to them in the triangle? For instance, is there something that is always true for right-angled triangles but is not true for the others?

How does this statement change if the triangle has an obtuse angle, or only acute angles?

If we make a square with 7 blue, 2 black and 2 red rods, and then take the 2 red and place them side-by-side with a blue one, the figure is no longer a square, but is the area of the new figure still equal to 81 sq. cms.?

Make squares using rods of many colors, e.g. 2 black, 3 yellow, 4 pink, 5 light green, 3 white, 2 dark green, 3 red. Take one or more rods away from their places and put them somewhere on the side of the rest so as to make a new figure. In this way you can produce a large number of arrangements which all look different but have one and the same area, equal here to 81 sq. cms.

This will give you an idea of the variety of polygonal figures obtainable from any one; in addition one property (here the area) remains unchanged.

Earlier on, we saw how the prism made of white cubes could give rise to prisms all of which had the same volume. If we go back to that question, instead of forming different prisms we could displace the white cubes in a variety of ways, always producing a three-dimensional body having the same volume.

We do not, of course, need to restrict ourselves to white cubes; any of the rods would do. If we make a three-dimensional

115

Modern Mathematics

structure and calculate its volume, the removal of any one rod from one place to another in the structure will give that structure another shape but will not change its volume.

Try several such structures. *

*

*

In Books II, III and VI of Mathematics, we worked out simple problems concerning the geometry of the rods. Here we have attempted to show that further manipulation of the rods will enable us to find deeper relationships. We have found invariants, and have attempted to remove the danger of forming direct but untrue relationships between perimeters and areas, or areas and volumes. We have also found that it is possible to open new vistas on families of figures (those, for example, having equal areas or volumes) which are needed for proper understanding of the calculus later on.

If, in the beginning of the study of geometry, these lessons on rectangular prisms obtainable from the rods are coupled with what can be learnt from the Geoboard (as described in Mathematics, Book VII), every child can be given an interesting foundation for geometrical experience, much more varied and holding much greater possibilities for the future than that based merely on measurements.

116

Conclusion

In this manual, we have avoided repetition of what can be found in either Mathematics (Books I to VII).

In order to help teachers to master the new presentation of the elementary stages of mathematics made possible by Cuisenaireâ&#x20AC;&#x2122;s invention of the colored rods, the first chapter has been devoted to a completely new treatment of elementary mathematical notions, requiring only attention and readiness to experiment on the part of the reader.

From that first chapter, the reader should acquire an attitude towards mathematics more akin to that of playing games with given rules than extracting rules from everyday experience.

Because the contents of that chapter refer more to the way of doing things than to the results obtained, it is called algebra. In view of the fact that this text is intended for teachers, the ideas selected are those that can serve in primary schools. We have no

117

Modern Mathematics

doubt that this chapter will be easily assimilated by all readers and will considerably increase their powers.

Chapter II is a birdâ&#x20AC;&#x2122;s-eye view of the content of the Mathematics books which offer a syllabus covering all the questions met by children in primary schools. It does not require comment because the foundations on which we have built quickly become obvious. Where questions have been left out, it is not that they are considered less important, but because they are treated in detail in other books. Teachers should refer to these in order to see whether their own answers and their own ideas exhaust as much or more of the mathematics involved in these fields.

Chapter III contains a short treatment of an aspect of mathematics which has not been extensively used in elementary teaching. As it is useful later on, and really easy, it has been included.

Teachers are wont to think that since arithmetic is needed in everyday life, this aspect of their work is one of the most important for their pupilâ&#x20AC;&#x2122;s future and for the transfer of learning. Nothing has been said in this book of that aspect of arithmetic teaching and this is dealt with in Books II, III and VI where the application of arithmetic in real life situations is explored extensively. The important thing here is the learning of mathematics and its meaning for teachers.

Is it not true that, after children have added 3 and 2 apples and found 5, 2 and 3 oranges and found 5, etc., teachers want them to know, and to be able to say, that 2 and 3 make 5 irrespective 118

Conclusion

of the objects to which they apply? So the aim is abstract mathematical knowledge and proficiency in those skills which can be acquired only by leaving out the objects. Does anyone actually add 1372 apples to 7239 in order to find the answer to the addition of these two numbers? Teachers want their pupils to increase their power over things by replacing actual actions by actions on paper which lead to the correct answer while saving time and energy. For this purpose, teachers use multiplication instead of repeated additions, rules instead of complicated operations, formulae instead of lengthy measurements.

With all this in mind, we substitute for the traditional methods the playing of games, which is, as we know, the spontaneous way in which children acquire much of their mastery over the environment. When the children have learnt how to play the game, we insert a symbolic use of what they have mastered, and they find it much easier that way. Is it more difficult for someone who knows what half any number or length is, to say that, in order to share 12 cards equally between two children, he must give 6 to each, than it would be for the child who always links numbers and objects? Our experience is to the contrary. If I know what

really is, I know it as a tool for working all

halves, and the half of 12 is merely a particular instance. There is an important difference between what we are saying here and the usual controversy between those who â&#x20AC;&#x2DC;generalizeâ&#x20AC;&#x2122; from the particular and those who particularize the rule. Our view-point (explained several times in other publications) is implicit in

119

Modern Mathematics

every page of this book: (i) we manipulate and watch what happens if we do this or that; (ii) we try to alter all that can be altered in the situations and what remains is our final acquisition; (iii) we formalize this and produce a notation for it.

Thus, although it looks abstract, it is not so, since it is intimately linked in our minds with all those variable situations. Then, (iv) we can pass on from the awareness of this experience to experience involving additional symbolic meaning for what enters into the operation. This final stage is that of applications and no child following such a course will get confused or ask the usual questions: ‘Have we got to add or multiply?’ or ‘Can’t you give us numbers?’ when presented with a so-called general question.

Because algebra puts powerful tools at the learner’s disposal, and because arithmetic gains deeper and more real meaning after algebra has been studied, we begin with algebra and remain constantly aware of it. Why should we first teach addition and then subtraction and then multiplication when they are all contained in every situation presented by the rods?

So we teach them all at once.

If we meet a multiplication like the one mentioned in the text, 6 × 7 × 8 = 336, why should we stop there? By traditional methods we try to justify doing so by applying it to situations where three numbers are multiplied together, as in the calculation of volumes, but with our approach this multiplication is seen as containing several other operations, and we form the table that expresses our awareness of the whole: 120

Modern Conclusion Mathematics

336 = 6 × 7 × 8 = 7 × 8 × 6 = . . . = 42 × 8 = 6 × 56 = 7 × 48 336 ÷ 6 = 56 × 336 = 56 etc.

336 ÷ 42 = 8 × 336 = 2 × 56 = 112

etc. × 336 = 7

In so doing, we become experts in the algebra of each situation (needed when we meet formulae in technical matters); we see so many numerical relationships that our experience of numbers is considerably widened and we increase the content of our memory and our store of facts that can be immediately recalled.

It becomes clear that, by thus extending the learner’s power we can offer him a wider curriculum and remove the many barriers created by earlier methods of teaching. How often have otherwise intelligent students been baffled by the inconsistencies they found in their mathematics course? How often have emotional disturbances caused by difficulties in assimilating arithmetic been responsible for the attributing of a low I.Q. to children?

If we have been able in this book to prepare teachers to read the more specifically methodological texts, written for the purpose of teaching rather than of helping the growth of teachers, we can look forward to the day when no child will be prevented by our mistakes from developing to the full his natural gifts.

This book is not complete, nor is the knowledge of the writer. Is there anyone who has worked with all children or who can give in a text (necessarily dated) all the answers to all concrete 121 121

Modern Conclusion Mathematics

problems? The appearance of this book will have been timely if after these many hours of work and play the reader has achieved a certain mastery of the use of the rods, gained some insight into mathematics, and recognized that if all children are to be educated, this new approach should be given its chance. The old methods discriminated between children and caused many to be labeled as non-mathematical when in reality they were only allergic to such methods. This being so, other methods must be found that will stimulate learning and present new challenges. It is our hope that the reading of this book will have convinced the reader that something new is in the air and is worth trying out.

Those who decide to do so will be helped by all the other texts mentioned in the bibliography, but much more by their own pupils. All I know has been learnt from the pupils and teachers I have met, as I said in the preface. Let that also be my conclusion.

122