The Cuisenaire Gattegno Method Of Teaching Mathematics A Course For Teachers Volume I

C. E. Chambers

Educational Solutions Worldwide Inc.

Published by Educational Explorers Limited in Reading, England. Set in ‘Monotype’ Bembo and printed in Great Britain by Lamport Gilbert Printers Limited, Reading First edition published in 1964. Reprinted in 1965. Reprinted in 1967. Reprinted in 2009. Copyright © 1986-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 0-00000-000-0 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

Table of Contents

Preface ........................................................................ 1 Author’s Note ..............................................................3 Chapter 1 The Cuisenaire Equipment: Outline Of The Course .........................................................................5 The Box Of Cuisenaire Numbers In Colour..................................... 5 Practical Experience With Cuisenaire Material ..............................8 References ........................................................................................9 Textbooks ....................................................................................... 13 Other Equipment ........................................................................... 16 Wallcharts, Product Cards And Counters...................................... 16 Geo-Boards..................................................................................... 18 The Aim Of This Book ....................................................................20

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method .....................................23 Operations And Processes .............................................................23 Ordinal And Cardinal Number ......................................................26 No Dependence On Counting ........................................................ 27 Understanding The Four Basic Operations ...................................28 The Basic Operations Are Seen To Be Interrelated And Can Therefore Be Taught Simultaneously ............................................30

Reverse Processes Can Be Seen; Every Situation Is Fully Exploited ........................................................................................................ 32 The Use Of Colour Names Allows For A Detailed Study Of Mathematical Concepts ................................................................. 33 Any Rod Can Be The Unit .............................................................. 35 Symbols For The 37 Basic Products Under 100 ............................36 Discarding The Rods ......................................................................38 Oral Discussion And Written Work............................................... 39 Horizontal Setting Out Of Written Work ......................................40 Automatic Response With Number Facts And Tables .................. 41

Chapter 3 Algebra Before Arithmetic Preliminary Exercises................................................................... 43 The Course In Pure Number.......................................................... 43 Free Play ......................................................................................... 45 The First Few Lessons—Classroom Organization ......................... 45 Packing ...........................................................................................46 Teaching Aims At This Stage ......................................................... 47 Special Games To Relate Colour With Rod Lengths ..................... 51 Directed Informal Activities .......................................................... 53 Patterns Which Provide A Background For Addition ...................58 Patterns Which Provide A Background For Subtraction .............. 59 Patterns Which Lead To Multiplication And Division .................. 61

Chapter 4 Algebra Before Arithmetic More Intensive Activities ................................................................... 65 Number Or Pre-Number Activities At This Stage? .......................66 Teaching Aims At This Stage ......................................................... 67

Factors Associated With The Oral Expression Of Mathematical Concepts .........................................................................................68 Factors Associated With The Written Expression Of Mathematical Concepts .........................................................................................69 The Place Of Oral Discussion In The Cuisenaire-Gattegno Method ........................................................................................................70 Colour-Name Notation ...................................................................71 Gattegno’s Exercises In Qualitative Arithmetic ............................ 73 Exercises Involving Addition ......................................................... 76 The Meaning Of ‘Plus’ .................................................................... 76 The Meaning Of ‘Equivalent’ ......................................................... 77 Introducing The Written Symbols For ‘Plus’ And ‘Equals’ ........... 78 The Meaning Of ‘Minus’................................................................ 80 Reading And Writing The Symbol For Multiplication .................. 81 Division ..........................................................................................82 Fractions.........................................................................................83 Reverse Processes ..........................................................................84 Exercises In Mixed Operations ......................................................85 Completing Patterns In Writing ....................................................86 The Part Played By Visual Images ................................................. 87

Chapter 5 The Study Of Numbers Up To 10 ............... 89 Application Of Gattegno’s Algebra Before Arithmetic Theory......89 Giving Numerical Value To The Cuisenaire Rods .........................90 Games To Relate Colour, Length And Numerical Value ...............93 Colour Names For Reference Purposes .........................................93 Staircases To Relate Colour, Length And Numerical Value..........94 Systematic Grading Of Number Is Not Essential ..........................96 Steps To Develop An Understanding Of Equality ......................... 97

Understanding Addition ................................................................99 Understanding Multiplication ..................................................... 100 Understanding Subtraction ......................................................... 100 Understanding Division................................................................101 The Four Operations With Numbers Up To 5 ............................. 103 The Use Of Brackets..................................................................... 104 Exploring The Number System ................................................... 107 Progression In Written Exercises ................................................ 109 Varying The Exercises................................................................... 111

Chapter 6 Teaching The Basic Concepts Of A Fraction ................................................................................. 113 Quotition And Partition In Division .............................................113 Quotition ....................................................................................... 115 Partition ........................................................................................116 The Study Of Fractions ................................................................. 117 Prerequisites For Work In Fractions ............................................ 117 Fractions As Relations Between Two Numbers ...........................118 Fractions With Numerators Greater Than One .......................... 120 Gattegno’s Exercises In ‘Mathematics’ .........................................121 Fractions As Operators ................................................................ 122 Equivalence Of Fractions ............................................................. 129

Chapter 7 The Study Of Numbers Up To 20 ..............133 Numbers May Be Studied In Any Order...................................... 134 Study Of A Particular Number..................................................... 135 Adding The Lengths Of A Handful Of Rods ................................ 138 Using Staircases For Exercises In Counting................................ 140

Developing The Concept Of Common Differences Between Pairs Of Numbers ...................................................................................141 Multiplication And Division (Including Fractions)..................... 146

Chapter 8 The Study Of Numbers Up To 100 ............ 155 A General Outline Of Activities Connected With Numbers Up To 100 ................................................................................................ 155 The Multiples Of Ten Up To 1oo .................................................. 156 The Four Operations With Rods End To End (Or Side By Side) .157 Carrying Out The Operations With Crossed Rods ...................... 159 Using The Rods To Represent Any Number Up To 100 ..............161 Studying A Selected Series Of Numbers ...................................... 162 Studying A Particular Number .................................................... 163 Discovering The Relationships Between The Numbers In A Particular Series ........................................................................... 164 Free Exploration Using Towers ................................................... 167 More Systematic Study Of Numbers Up To 1oo .......................... 168 Manipulating The Rods For The Four Operations ....................... 171

Chapter 9 Gaining Automatic Response With Number Facts And Tables.......................................................177 Automatic Response May Be Gained Concurrently With The Development Of Knowledge ........................................................ 178 Gaining Automatic Response With Tables By Means Of The Cuisenaire Rods ........................................................................... 180 Gaining Automatic Response By Means Of The Wallchart And The Product Cards And Counters ....................................................... 183 Gaining Automatic Response By Means Of Traditional Methods .......................................................................................................191

Chapter 10 The Study Of Numbers Up To 1000—I .... 197 Exploring The Number System ................................................... 197 Learning The Fixed Procedures................................................... 198 The Study Of Numbers Up To 1ooo ............................................. 199 A Plan For The Discussion Of Numbers Up To 1ooo ................. 200 The Notation For Numbers Up To 1ooo Which Are Multiples Of 1oo ................................................................................................ 201 Multiplying (And Dividing) By 1oo And By 1o ............................204 The Notation For Numbers Which Are Multiples Of 10 .............204 Numbers Up To 1ooo Which Contain A Units Figure ................. 210 Some Special Exercises In The Study Of Numbers Up To 1ooo.. 218

Chapter 11 The Study Of Numbers Up To 1000—Ii ....221 Exploring Numbers Of Three Figures Which Are Not Multiples Of 10 ..................................................................................................222 Discovering Useful Products........................................................223 Doubling And Halving Numbers Up To 1ooo..............................224 Multiplying And Dividing By 2 ....................................................226 Multiplying By 5 ........................................................................... 227 Multiplying By 4, 8, 16, 32 And 64 ..............................................228 Multiplying By 20, 40, 80 And By 50 ..........................................229 Multiplication By 25.....................................................................230 Making Use Of The Ability To Double......................................... 231 Trebling Up To 1ooo And Dividing Numbers By 3...................... 231 Using Numbers In Any Series Of Trebles For The Study Of Multiplication And Division ........................................................232 Studying Remainders In Division................................................233 Applying The Ability To Treble To Multiplications By 6 And 9.. 233

Multiplying By 30, 60 And 90 .....................................................234 Using Doubling And Trebling For Rapid Multiplications...........234 Multiplying By 7 ........................................................................... 235 Multiplication By II......................................................................236 Multiplication By 12 .....................................................................236 Multiplying Two-Figure Numbers Together ...............................236 Other Means Of Finding Useful Products ...................................238 Some Special Exercises In The Manipulation Of Squares .......... 247 The Reading And Writing Of Numbers .......................................249

Chapter 12 Discovering Systematic Procedures ....... 253 Horizontal And Vertical Notation................................................254 Discovering The Fixed Procedure For Addition .......................... 257 Discovering A Procedure For Subtraction...................................263 Discovering The Procedure For Multiplication ........................... 272 Discovering The Procedure For Division..................................... 277

Some Publications On The Cuisinaire Gattegno Method ................................................................................285

Preface

This Course by Mr. C. E. Chambers may prove of value not only to teachers in New South Wales, where the author lives and works, but to teachers outside as well. For this reason some alterations have been made to the original manuscript in order to eliminate elements that might otherwise not have been understood by readers, or which might have made the text less fluent. This interference we hope has not diminished the value of the work. Mr. Chambersâ€™ enthusiasm is obvious. It is also clear that he has worked hard in order to assimilate the ideas he attempts to simplify and clarify in this work for the benefit of his colleagues who took this course. One particularly valuable aspect of the text is that the author is simultaneously trying to understand new ideas while old ones are still operating in him, and trying to eliminate those he feels block the light the new ones shed. As a practicing teacher, Mr. Chambers has shown courage and determination in taking upon himself the considerable task of 1

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

communicating an unusual, and not always easy, philosophy of teaching elementary mathematics to perhaps skeptical and not yet informed colleagues. In addition, his devotion to education becomes apparent in several places when he demands that practice in classrooms should decide whether a suggestion has merit or is merely fanciful. For his contribution to the tasks of teaching, he deserves the gratitude of his colleagues. I personally am pleased to be able to present his work to the English-speaking teachers of the world, and of Australia in particular. C. GATTEGNO General Editor

2

Authorâ€™s Note

Towards the end of 1961, I was commissioned by the New South Wales Department of Education to write a series of twenty lectures on Cuisenaire-Gattegno Arithmetic for use as an in-service correspondence course for teachers in the schools of New South Wales. These were distributed to interested teachers during 1962 and 1963. During 1963, the original lectures were revised and amended to conform with the ideas that I had developed as a result of increased practical experience with the Cuisenaire equipment and Dr. Gattegnoâ€™s revolutionary theories. This revised correspondence course will be available to New South Wales teachers in 1964. As the contents of this volume are based on my original lectures, plus the later revisions and amendments, and as some special adjustments have been made by Dr. Gattegno himself to allow my ideas to be presented more suitably in book form, I gratefully acknowledge first, the kind permission of the New

Â

3

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

South Wales Department of Education for allowing me to publish material which is Crown Copyright and second, the practical assistance of Dr. C. Gattegno and his colleagues in Educational Explorers in converting my lectures so satisfactorily into book form. C. E. CHAMBERS December, 1963

4

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

Considerable interest in the problem of teaching arithmetic has been aroused by the comparatively recent appearance of certain materials, used originally by Catherine Stern of the United States on the one hand and by Georges Cuisenaire of Belgium on the other. The practical techniques of the Cuisenaire method are still imperfectly known and it is the purpose of this book to reveal some of the possibilities of this equipment when it is properly used.

The Box Of Cuisenaire Numbers In Color For purpose of reference and for practice in manipulation, one must have access to a box of Cuisenaire rods—‘Numbers in

5

The Cuisenaire Gattegno Method Of Teaching Mathematics

Color’ is the correct term. Each box consists of 291 colored rods from one to ten centimeters in length, and all of section one square centimeter. Rods of the same length are the same color and there are ten colors. The total length of the rods of one color, that is when all rods of the one color are placed end to end, is either exactly, or just less than, one meter; thus: white red light green pink yellow

100 × 1 cm. 0 × 2 cm. 33 × 3 cm. 25 × 4 cm. 20 × 5 cm.

dark green black tan blue orange

16 × 6 cm. 14 × 7 cm. 12 × 8 cm. 11 × 9 cm. 10 × 10 cm.

The significance of these colors will be realized later but it can be seen immediately that they add to the attractiveness of the equipment and thus appeal to children; they are a means of reference for identification and assist in the association of number with size. It should be noted that certain of these colors are interrelated and form color families; this is intentional. Red, pink and tan all have a red pigmentation; light green, dark green and blue all have a blue pigmentation; yellow and orange have a yellow pigmentation and

6

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

FIG. 1. One rod of each length. then there are the black and the white. Numbers and their multiples can thus be represented by these color relations.

Â

7

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 2. Families of related colors. The colored rods are the basic foundation for the method. They were originally developed and successfully used by M. Georges Cuisenaire, a Belgian, while detailed suggestions for their use have been expounded and published by Dr. Caleb Gattegno.

Practical Experience With Cuisenaire Material At some time while studying this course, preferably after having gained experience with the rods, the reader should introduce them to his class. He might do it for some specific purpose, perhaps to teach one simple item in arithmetic, and test the pupils’ reactions. His aim should be to gain some practical experience immediately to assist in decisions concerning the fuller use of the equipment later.

8

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

This recommendation for a restricted use of the rods need not apply to kindergarten or first grade where the full Cuisenaire method could be introduced even as late as the middle of the school year. Among those keenly interested in the method, the feeling seems to be growing that the most practical way of introducing this equipment into schools is at the kindergarten or first grade levels, gradually extending it each successive year to higher grades. The rods can however be satisfactorily introduced at any stage in oneâ€™s schooling. As these discussions proceed, it will become clear 1

that the Cuisenaire rods can be used incidentally as a teaching aid or prop for the teaching of mathematics as outlined in our present curriculum,

2 that the rods can be used as the basis for an entirely new and different curriculum.

References A second recommendation is that some of the following books be read as soon as possible, perhaps in the order listed. In them, Dr. Gattegno, chiefly, elaborates in some detail on the use of the Cuisenaire rods and other equipment.

9

The Cuisenaire Gattegno Method Of Teaching Mathematics

C. Gattegno, A Teacher’s Introduction to the Cuisenaire-Gattegno Method of Teaching Arithmetic, Educational Explorers, Reading, 1960—reprinted several times. This was written especially for primary school teachers and was published later than the first edition of the books described below. Gattegno says in the Introduction to this book: ‘The teaching tradition, built up over centuries, consists, in large measure, of a combination of counting skill and memorization. Our purpose in the new method is not to do away with counting but to give it its proper place beside the other activities. When teachers generally discover the advantages of this novel approach, and come to grips with the deeper principles that underlie it, the results will be truly remarkable’. G. Cuisenaire and C. Gattegno, Numbers in Color—A new method of teaching processes of arithmetic to all levels of the Primary School, Heinemann. First published, 1954; second edition, 1955; third edition, 1957; reprinted (twice) 1958 and 1960. This book should be read in conjunction with A Teacher’s Introduction above. It is in two parts, the sections written by Gattegno describe ‘some of the uses to which the material can be put in classroom situations at the infants, primary and secondary levels’, (preface to the first edition); and those written by Cuisenaire are an adaptation of Les Nombres en Couleurs written in French for Belgian teachers and describe ‘the means by which the child can pass easily and with certainty from the stage of observation (seeing, touching, feeling) to that of firm

10

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

consolidation at the concrete level, as a preliminary to abstraction and subconscious automatism’ (Introduction to Part 2). C. Gattegno, From Actions to Operations—A detailed outline of mathematics teaching, 1958; (out of print). This book was originally written with deaf children in mind; so it is ‘a deliberate attempt to displace the stress from a teacher-centered type of lesson’ (through the medium of language) ‘to a learner-centered one’, through the medium of the Cuisenaire colored rods for arithmetic and Gattegno’s own geo-boards for geometry. The successive phases in the use of the colored rods and of the nailed boards are treated from this point of view and in such detail as to be of interest to all teachers, not only those of deaf children. C. Gattegno, Modern Mathematics with Numbers in Color—a manual for primary school teachers, Educational Explorers, Reading, 1959; revised edition 1963. In this manual Gattegno gives his ideas on the classroom use of the Cuisenaire rods, ideas that have matured and crystallized after ‘five years’ intensive and extensive investigations in very varying circumstances’ (Preface). He presents to the teacher of primary classes an ‘Algebra before Arithmetic’ course in mathematics (something totally different from Cuisenaire’s own method). The manual is intended ‘not as a guide to teaching but as a complete course of learning, first by the teacher and,

11

The Cuisenaire Gattegno Method Of Teaching Mathematics

through his understanding of the subject matter, by the pupil’ (Preface). C. Gattegno, Now Johnny Can Do Arithmetic, Educational Explorers, Reading, 1961; revised edition 1963. This book has been specially written for the parent who wishes to assist his child in the study of arithmetic. Because the emphasis is on the child’s own problems the text is such that it can best be used in the home when time for play is plentiful, and yet its clear exposition of many of the uses of the Cuisenaire rods may also be of much interest to the teacher. The book caters for the child who has not yet been subjected to the demands soon to be made of him and also for the child who has already lost his way in his mathematical studies. This is done by suggesting games which introduce fundamental mathematical concepts and which also can serve as remedial exercises for the older pupil. In addition to these publications Dr. Gattegno has also designed and made a film strip called ‘Numbers in Color’ in which the use of the Cuisenaire material is illustrated. When used in conjunction with the Teacher’s Notes that accompany it, it can be helpful, especially when viewed with some rods handy for manipulation if needed. By this time the reader will have perhaps realized that while Georges Cuisenaire, the teacher, invented and successfully used the equipment, it was Caleb Gattegno,

12

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

psychologist, scientist and mathematician, who more fully realized its potentialities, and energetically encouraged its use throughout the schools of the world. While Gattegno repeatedly acknowledges that Cuisenaire was the source and first inspiration of this revolution in teaching arithmetic, he himself deserves credit because he has left no stone unturned in his efforts to promulgate the method.

Textbooks Much more important is the series of books, which in the 1963 edition is called Mathematics with Numbers in Color, Educational Explorers (first published from 1957 to 1961). In these books, Gattegno has set out a graded series of exercises and activities as a guide for teachers and for use by pupils who have the equipment in their classrooms. A thoughtful analysis of these exercises reveals how Gattegno would put his theories into practice. There is no indication in the books of the purpose underlying any exercises so that it becomes essential, before they can be used effectively, for teachers to become conversant with the general principles and practical details of the method. These books will be referred to constantly in this course and it is recommended that most, if not all of them, be obtained.

BOOK I

Â

13

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

I

Sessions of free play

II

Qualitative work with the rods

III

Literal work

IV

Number work. Measure. Study of numbers up to 10

V

Applications

VI

Study of numbers up to 20

BOOK II I

Study of numbers from 1 to 100

II

Vertical notation

III

The clock

IV

Numbers up to 1000

V

Days and weeks

VI

Reading and writing numbers

VII

Procedures, operations

VIII

Grouping, cost price, selling price, profit

IX

Perimeters, areas, volumes

algorithms

BOOK III

Â

I

Money

II

Unitary method

III

Length

IV

Area and volume

V

Capacity and weight

VI

Decimalisation of money

14

for

the

four

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

BOOK IV I

Fractions as operators

II

Study of fractions

III

Study of decimal fractions

IV

Percentages

BOOK V I

The formation of numbers

II

Different bases of numeration

III

Divisibility and prime numbers

IV

H.C.F. and L.C.M.

V

Squares, cubes and square roots

VI

The set of integers

BOOK VI I

Length, area and volume

II

Capacity and volume of round bodies

III

Weight, mass and density

IV

Measures, their formalisation

V

The C.G.S., M.K.S. and M.T.S. systems of units

VI

Proportions and mixtures

VII

Problems depending on measurements

VIII

Problems on speed

IX

The perpetual calendar

BOOK VII

Â

15

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

I

Simultaneous equations; about directed numbers

II

Permutations and combinations

III

Sets and subsets ; algebra of sets

IV

Arithmetic progressions and geometrical progressions

V

The geometry of the geo-boards

Other Equipment Before proceeding to outline the subject matter of the respective chapters in this course, one must mention the remaining items of equipment connected with the Cuisenaire-Gattegno method. These might be regarded as subsidiaries.

Wallcharts, Product Cards And Counters The wallchart and product cards are concerned with the symbolic representation in color of the 37 products under 100 that have factors of ten or less. The organization of the multiplication facts into 37 products is especially interesting to teachers who know the difficulties associated with teaching the multiplication tables. The promise of economy in teaching time (and in learning time from the point of view of the pupils) that seems possible from this organization is an attractive feature of the method.

Â

16

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

Each product is represented by means of colored crescents representing the factors, placed on opposite sides of an empty white circle in which the product may, when first introduced, be lightly written in pencil (Fig. 3). The purpose of the empty circles is for circular counters, on which the product is printed, to be inserted (Fig. 4). At the same time, there is always opportunity for the products to be provided mentally (and this is the purpose of most games played with them) from an observation of the colors of the crescents, which correspond, of

course, to the colors of the rods; it will be seen that the rods may also be used to represent a product (Fig. 5). FIG. 3.

FIG. 4.

FIG. 5.

The wallchart contains all the 37 products arranged horizontally in a particular order, usually by doubling. The arrangement is set out below:

Â

17

The Cuisenaire Gattegno Method Of Teaching Mathematics 4

8 10 6 9

16 20 12 18

25 15 14 63

32 40 24 36

50 30 28 49 45 27 35 21

64 80 48 72

100 60 56 81 90 54 70 42

The product cards are 37 separate cards forming a full pack of the products symbolized in color. They can be used in a number of games on products, factors and fractions. Further details concerning the part each of these items of equipment plays in the method will be discussed in chapter 9.

Geo-Boards The geo-boards are Dr. Gattegno’s invention, designed as an aid in the teaching of geometry. They consist of boards into which nails have been driven to allow for the making of a wide variety of geometrical shapes with colored elastic bands. The method of teaching geometry by means of these geometry boards is parallel in principle to the method of teaching arithmetic, that is, discovery by the pupil, during a wide variety of experiences through the medium of this concrete material, of basic geometrical ideas associated not only with the elementary properties of plane figures but also with spatial relationships generally.

18

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

The full set consists of a 9-pin, a 16-pin and a 25-pin board, an octagon, decagon, dodecagon and double hexagon. Details of their use, together with the use of Cuisenaire rods for the study of perimeter, area and volume, will be considered in Chapters 18 and 19. The 25-pin board, for example, is 30 cm. square. A nail is driven into the centre of each of the 25 squares which are scored on the base, each of which is 6 cm. square. The triangles shown in Fig. 6b below are made by colored elastic bands and stand out clearly on the flat natural wood base

FIG. 6a.

Â

FIG. 6b.

19

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

The Aim Of This Book The scheme of this book is not designed to follow any primary curriculum in mathematics but rather to present a new, almost revolutionary curriculum, a complete course in pure arithmetic that affects conventional methods both directly and indirectly. The teachers concerned should understand it from beginning to end, at least in general principles. Each chapter of this book, although detailed for its particular purpose, must be closely related to the next. The ground now seems ready for the discussion of Cuisenaire activities in the classroom. After the next chapter, which outlines in general terms the main features of the method, a series of six chapters will deal with what has been described as a revolutionary departure from conventional methods of introducing arithmetic and teaching the various techniques associated with the basic operations. Chapters 3 and 4 explain how arithmetic can be introduced and many mathematical ideas demonstrated without any reference to number at all, reference being made entirely to the colors of the rods. After this discussion of the algebraic approach to the introduction of arithmetic, chapters 5, 7 and 8 will describe detailed activities which allow pupils to discover for themselves and gradually consolidate their ideas concerning the number system and the four basic operations. Chapter 6 demonstrates how fractions can be taught to infant and lower primary grades.

Â

20

Chapter 1 The Cuisenaire Equipment: Outline Of The Course

After this period of informal study of basic mathematical concepts pupils pass through preparations for an understanding of the formal processes (Chapter 12), necessary for exercises with large numbers (studied in Volume II), by becoming familiar with numbers up to 1000 (Chapters 10 and 11) while at the same time gaining automatic response with the basic number facts and tables (Chapter 9). In Volume II of this work, Chapter 13 (the study of large numbers) completes the course in the study of pure number. The remaining chapters deal with the use of Cuisenaire rods for teaching fractions (Chapter 14), decimal fractions and percentages (Chapter 15), English money and measures (Chapter 16), problems (Chapter 17) and geometry (Chapter 18). The use of Gattegnoâ€™s geo-boards are discussed in Chapter 19, and a recapitulation of the course and a final assessment of the method is made in Chapter 20.

Â

21

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

Operations And Processes The difference between traditional methods of teaching arithmetic and those recommended by Gattegno is bound up in the meaning of the words ‘operation’ and ‘process’. Throughout these discussions each word will be given a specific meaning. Any teacher who has used the Cuisenaire rods realizes that pupils learn to add, multiply, subtract and divide (either with or without the rods) without employing any specific method that might be recognized as a process. The word ‘operation’ will therefore be used with this particular connotation. Finding an answer by means of any unspecified method will be regarded as an ‘operation’.

23

The Cuisenaire Gattegno Method Of Teaching Mathematics

As a result, the word ‘process’ must refer to the fixed and conventional procedures which we use when dealing with large numbers. It involves such procedures as ‘borrowing’ and ‘paying back’ in subtractions, and ‘carrying’ in the other operations—all those matters implied under the heading of ‘mechanical arithmetic’. In many schools at the present time, mechanical arithmetic is introduced almost as soon as the child has mastered the minimum of number facts required. He learns to count and proceeds almost immediately to addition, followed in turn by subtraction, multiplication and, finally, division. His success and speed are usually relative to his memory of the number facts involved in the particular process. This approach makes co-ordination throughout the various grades important, so that the same process is maintained from grade to grade. With this end in view many teachers have searched for years, and changed their minds often, in an effort to find the simplest and easiest process to follow for each type of sum. As a result, a large proportion of the teaching time in arithmetic, from infant grades onwards, is spent drilling number facts for use in a process which in turn must also be drilled until it becomes mechanical. The end result, furthermore, is that the child knows only one process for each of the basic operations. The Cuisenaire-Gattegno approach is fundamentally different and may be summarized as follows:

24

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

After a period of free play to develop familiarity with the rods and skill in handling them, the infant pupil proceeds with a wide variety of pre-number experiences to develop the concepts necessary for an understanding of arithmetic. Then follows, possibly in first but certainly in second grade, a study of numbers up to 10 and beyond, which involves activities in exploring the number system. Situations are created with the rods in which children discover for themselves a variety of ways of carrying out the four operations. This understanding of the number system is developed further in the third grade; numbers up to 1000 are studied and for the first time in the course, an attempt is made to secure mastery of the number facts and tables essential for following fixed procedures. Many of these may already be known, having been learnt incidentally. As the need becomes more and more apparent to the third grade pupil dealing with larger numbers, he realizes that the number facts must now be clinched. This, then, is the main feature of the Cuisenaire-Gattegno course in pure arithmetic, made attractive and interesting by the use of ‘Numbers in Color’. Exercises in applied arithmetic, in mathematical situations arising from the pupil’s social experiences, in problems relating to money and other measures derived from social needs—are, says Gattegno, basically dependent on an understanding of pure arithmetic. They consist in recognizing the specific situation described in the problem, doing the arithmetic that is involved in finding the answer, and then expressing the answer in terms of that situation.

25

The Cuisenaire Gattegno Method Of Teaching Mathematics

Â

Ordinal And Cardinal Number One must warn readers that the Cuisenaire rods do not provide the answers to all problems in the teaching of arithmetic. For ex ample, it is better to continue to use pins, apples, trees or objects other than the rods for the teaching of ordinal number, which is number in sequence, such as is used in counting. On the other hand, the Cuisenaire rods are the best, if not the only means yet discovered of allowing children to gain quickly a clear concept of cardinal number, which is number as an entity, as a whole. It has always been difficult for young children to distinguish clearly between 1

the last object counted as five and

2

the whole group of objects as five

yet the Cuisenaire rods demonstrate this quite simply. If the white rod is valued at one, and five white rods equal a yellow rod, then the yellow rod stands for five. With the rods set out as in Fig. 7, the child can see five not only in relation to four plus one more, but also as an entity, a whole, in which there is no first, second . . . fifth object.

Â

26

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

FIG. 7. The development of this idea of cardinal number is possible chiefly because of the absence of marks on the rods.

No Dependence On Counting Using counters, the child would find the sum of 3 and 5 by counting out three objects, and then five objects, putting them together, and then counting the total; or by counting three and counting on five more. However, it is doubtful whether the child can see with counters that, in reality, he is bringing together two groups, or wholes, which equal or create a third group, or whole. Using the rods, on the other hand, he can see every time that it is groups he is handling, a fact which removes his dependence on counting and hence on the use of fingers or nods of the head. From this point of view alone the unmarked Cuisenaire rods can be responsible for a considerable advance in teaching methods.

Â

27

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 8. A whole ‘three’ (light green) plus a whole ‘five’ (yellow) creates a whole ‘eight’ (tan).

Understanding The Four Basic Operations Because there are no distracting marks on the rods and because children can see numbers as wholes, they can be shown the real meaning of addition, subtraction, multiplication and division. By placing the red and light green rods end to end and measuring them with a yellow they can see that the two groups (the whole ‘two’ plus the whole ‘three’) together create a third group (the whole ‘five’).

FIG. 9. 2 + 3 = 5 (measured by the white rod). Moreover, in the operation of subtraction they can see that they do the reverse, that is, break the whole ‘five’ down into two parts, also seen as wholes.

28

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

FIG. 10.

5–2=3

For multiplication, Fig. 11 demonstrates that three groups of the same size (three whole ‘threes’) equal a third group (a whole ‘nine’).

FIG. 11. The reverse of multiplication can be seen as division; for example, how many groups called ‘three’ would equal a group called ‘nine’? This would be a matter of placing the blue rod down first, and then measuring it with light green rods. Finally, the relationship between any pair of rods can easily be seen as a fraction. In Fig. 12, for example, the red rod is seen as half the pink or as a quarter of the tan.

29

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 12a.

FIG. 12b

Indeed, the Cuisenaire rods allow for the very early introduction of the fraction concept, a matter traditionally regarded as difficult and therefore normally left until quite late in primary schools. Such then is the simplicity of the Cuisenaire rods that the child can easily discover for himself the various number relationships associated with addition, subtraction, multiplication and division. To use a ‘more difficult’ example, a second grade pupil who knows how to set out rods for numbers up to 100 can discover the relation in division between two numbers such as 19 and 40, usually expressed as ‘How many 19’s in 40?’ Placing alongside four orange rods end to end two orange-plus-blue lengths, he sees that the remaining space can be filled with a red rod.

The Basic Operations Are Seen To Be Interrelated And Can Therefore Be Taught Simultaneously From an observation of the illustrations above, the close relationships between addition, subtraction, multiplication and

30

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

division may have been noticed. The next illustration will clarify this further.

FIG. 13. The relationships between the operations can be seen in the following statements based on the above pattern: 2+2+2=6 3×2=6 6–2=2×2 6÷2=3

6 – (3 × 2) = 0 6–2–2–2=0

When the operations are understood, the child soon realizes that the basic difference between addition and subtraction on the one hand, and multiplication and division on the other, is that the latter are concerned with groups of magnitudes of equal size. Here the rods almost demand that the four operations be taught simultaneously. No longer is it necessary to concentrate on one operation or process at a time or on any particular difficulty in that operation or process. This also explains why Cuisenaire graded activities from small to larger numbers.

31

The Cuisenaire Gattegno Method Of Teaching Mathematics

Reverse Processes Can Be Seen; Every Situation Is Fully Exploited When one particular number fact or relationship has been given, the Cuisenaire rods allow for the reverse situations to be easily demonstrated and therefore taught simultaneously. For e× ample, observing the rods as set out below, the child can read and understand the following relationships:

FIG. 14.

4+5=9 5+4=9 9–5=4 9–4=5 Or, for multiplication and division:

FIG. 15. 2 + 2 + 2 = 6 3×2=6 3+3=6

6÷2=3 6÷3=2 ×6=3

32

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

2×3=6

×6=2

From just one pattern made with the rods, a number of situations can be exploited fully. Only the simplest facts have been listed in Figs. 14 and 15, but it is surprising how, after a relatively short experience with such exercises, pupils looking at the pattern in Fig. 15, for example, could read or write such expressions as ×6=6–3 (

× 6) + 2 = 6

6 – (2 × 2) = (

× 6) + (

2×3–(

×6 × 6) + (

× 6) = 6

× 6) = 3

The Use Of Color Names Allows For A Detailed Study Of Mathematical Concepts Gattegno calls this ‘qualitative arithmetic’ and uses the rod colors as the basis for his ‘algebra before arithmetic’ method of studying a wide variety of mathematical concepts before the child has any specific number knowledge. The child can express any one of the four operations using names-names. For example, he can read from the pattern in Fig. 16, that the light green rod and the pink rod are together equal to the black rod, and write g + p = b.

33

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 16.

FIG. 17. The yellow rod plus the pink rod equals the pink rod plus the yellow; or y + p = p + y Thirdly, because equivalent additions and common differences can be illustrated so easily with Cuisenaire rods and understood so readily by the child, they are given much more emphasis than has been possible in the past.

FIG. 18. Equivalent additions. FIG. 19. Common differences. Many other mathematical concepts can be illustrated with the colored rods, such as the meaning of plus, minus, equals,

Â

34

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

products, factors, squares and cubes. Sufficient has been said for the time being, however, to indicate that color, like the absence of marks on the rods, provides an opportunity for considerable progress in the teaching of mathematics.

Any Rod Can Be The Unit The Cuisenaire rods, as semi-abstract materials, may symbolise many things ; during periods of free play pupils use them for bridges, boats or trains. In arithmetic, because there are no markings, any rod can be chosen as the unit. If the white is one, then the orange is ten; if the yellow is one, then the orange is two; but if the orange is one, then the yellow is a half, and the white is one tenth. No rod has any value in itselfâ€”it is simply a piece of wood; its value is derived entirely from its relationship to the rod which is taken as the unit. The implications of this fact are obvious to any teacher who can see the representational use of the rods for problems dealing with money and other measures.

Â

35

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 20.

= 2 × 6d. = 6d. + 2 × 3d. = 9d. + (3 × 1d.)

Three-eighths of a gallon tin of petrol has been used for the motor mower. How many pints of petrol are left in the tin?

∴5 pints are left over FIG. 21.

Symbols For The 37 Basic Products Under 100 Products can by symbolized by crossing the two rods which represent the factors. For instance:

36

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

FIG. 22.

15 = 3 × 5 = 5 × 3 42 = 6 × 7 = 7 × 6

FIG.

23

The basis for this particular symbolism is as follows, demonstrated with 3 × 5 = 15. 1

First, the following pattern is made:

FIG. 24. This clearly shows that 15 (or 10 + 5) = 3 × 5 = 5 × 3. 2

Second, this equivalence is again shown by 1

making a rectangle of the three yellow rods and

37

The Cuisenaire Gattegno Method Of Teaching Mathematics

2 making a rectangle of the five light green rods.

FIG. 25a

3×5

FIG. 25b.

5×3

3 It will be seen from these illustrations that the two rectangles are congruent, but, to demonstrate this concretely, the rectangle of yellow rods can be placed on top of the rectangle of light green rods. 4 To symbolize 15 as the product of 5 and 3, there is need to use one representative rod from each rectangle only, crossed at right angles as in Fig. 22 above, thus giving the total area of the rectangle, or the number of rods of one color needed to form it.

Discarding The Rods The child should be allowed to use the rods for as long as he needs them. At no stage is pressure put on him to discard them. In practice, it will be found that he will do this automatically; he will dispense with them when he understands the situation involved. The multi-sensory appeal of the Cuisenaire rods hastens this process, reduces the rate of forgetting and allows

38

Chapter 2 Some Important Features Of The Cuisenaire-Gattegno Method

the observant teacher to gain much valuable information about the childâ€™s progress.

Oral Discussion And Written Work Oral discussion of a mathematical situation set out with the Cuisenaire rods invariably precedes written work. The use of the correct mathematical terms such as plus, minus and equals from the very beginning of the childâ€™s course in arithmetic, is recommended. The child reads what he sees in any particular pattern of rods, closes his eyes and says it out loud again; this helps to create images that will remain firmly in his mind. Where there is no difficulty in the mechanics of writing, these oral exercises can be repeated in writing. A first grade child who can read from his rod pattern that two plus three equals five should also, as soon as he can write, use the written symbols 2 + 3 = 5. Ordinarily, the written work consists of 1

writing what can be seen in a particular pattern of rods, for example 3 + 6 = 9;

2 completing patterns in writing (for example 3 + 6 =) 1

Â

first by making the particular pattern with the rods and thus finding the answer;

39

The Cuisenaire Gattegno Method Of Teaching Mathematics

2 later, when the particular number situation is understood and known, by writing the answer direct, without recourse to the rods at all.

Horizontal Setting Out Of Written Work For the informal study of the number system the horizontal notation, ‘the language form of the sum’ as Gattegno calls it, is to be preferred. For example, with 24 + 36 + 19 = , the child is encouraged to evolve his own method. In this case, a particular child may decide to add the tens first, reaching 60; then the 4 and 6 to make an extra 10 (= 70); and finally to add the 9 (= 79). Although it may not be as efficient as the formal method used with the vertical notation it helps build a foundation for the introduction of formal techniques—the primary aim throughout the early phases of the Cuisenaire-Gattegno course. All these features listed so far and briefly discussed are concerned chiefly with the early phases of the course; in addition, most illustrations and diagrams have been purposely restricted to numbers below ten and to single rods. This has been done first because it has been assumed that this is the reader’s introduction to the use of the materials and second because all the features outlined will be frequently illustrated in the detailed classroom activities to be discussed in later chapters, particularly those features pertaining to primary grades.

40

The Cuisenaire Gattegno Method Chapter 2 Of Teaching Mathematics Some Important Features Of The Cuisenaire-Gattegno Method

A few matters concerning the later phases of the course remain to be mentioned.

Automatic Response With Number Facts And Tables Throughout this preliminary period, that is, until a thorough understanding of the number system has been achieved, no direct attempt is made to gain automatic response with number facts and tables. In this respect, Dr. Gattegno argues that there is little need for drill at any stage because mastery of these number facts is always gained incidentally, either through repeated manipulation of the rods or by means of the games and exercises with the wallchart and product cards. This theory has still to be proved by every teacher for himself but, whether gained incidentally or by direct pressure these number facts must be mastered before fixed procedures are introduced.

39 41

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

The Course In Pure Number The unity of this course in arithmetic can be emphasized by setting out the major stages and the aims at each stage. 1

Free Play

Aim: to develop familiarity with the rods and skill in handling them. 2

Pre-number

Â

43

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

Aim: to develop the necessary mathematical concepts essential for an understanding of arithmetic. 3

Experience with numbers 1-10

Aim: to introduce the child to number and to develop his understanding of number. 4 Experience with numbers over 10 Aim: to give the child a wide experience of number and an understanding of the functioning of the number system. 5

Mastery of number facts

Aim: to teach those number facts and tables essential for the effective use of the processes. 6 Processes Let it be noted once again that the Cuisenaire-Gattegno method of teaching arithmetic is one entity, a course in itself, and that the rods are not just another teaching aid; this point is being stressed in the hope that the reader will clearly understand the foundation on which the piecemeal discussions in the following chapters are based. Given this general plan, he should see the detailed pattern unfold itself as each successive chapter is studied.

Â

44

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

Free Play In Book I of Gattegno’s Mathematics with Numbers in Color (from now on noted only as Mathematics), the first part is devoted to Sessions of Free Play. This consists of two perfectly blank pages. He aims, of course, to emphasize that ‘free play is, precisely, free’ (Teacher’s Introduction p. 29). This is certainly true but, for the teacher introducing Cuisenaire rods for the first time, a little special guidance may be helpful.

The First Few Lessons—Classroom Organization One box of rods for every four pupils is usually sufficient for most purposes, and if the same box is allocated to each group each lesson, pupils come to take a proprietary interest in the material and usually are more careful. Tables, where the rods can be accessible to each pupil in the group, are more suitable than desks. The rods should be jumbled together to give the appearance of abundance, but during later stages of the course it may be found preferable to leave the rods in their respective compartments in the box.* * In classrooms with fixed dual desks perhaps the best solution is for the two pupils in one desk

to hand over half (or approximately half) the contents of each compartment to the pair sitting in front. If extra boxes can be obtained, half the normal supply to each dual desk works satisfactorily, for example with a separate half-box to each desk.

45

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

These practical problems are all a little unsteadying for a start but soon become routine in the hands of a teacher anxious to experiment with the method.

Packing The second matter is the important one of packing the rods away at the end of each lesson. For the first few weeks this should be made a definite part of the lesson. Up to 20 minutes should be allowed for the first fortnight or so, including the time spent by the teacher inspecting the packed boxes. The routine of systematic inspection each day will ensure that pupils are careful with the equipment. After a time periodical checking only, perhaps once a week, will prove sufficient and the total time allowed for packing can usually be reduced to a few minutes. Packing the rods by the children is a valuable social experience. The task should be shared between all members of the group; one child for example, might pack the red family of colors (red, pink and tan); another the blue family (light green, dark green and blue); a third the yellow and orange; and the fourth, the white and black. An alternative method would be to pack in order of size: (a) the white, (b) the red and light green; (c) the pink, yellow and dark green; (d) the black, tan, blue and orange rods. The larger rods are easier to pack than the smaller ones.

46 Â

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

Â The same organization could be held to for a week or so and then rotated so that ultimately each child will have gained experience in packing each color. A further alternative of course, and one which develops naturally after a few months, is for each group to make its own decisions on the matter without any particular direction from the teacher. The important point is that some definite arrangement of the rods, either by size or by related colors or any other way suitable to the teacher, not only makes pupils methodical but, when all boxes are packed in the same way, makes checking easier.

Teaching Aims At This Stage A decision to introduce the Cuisenaire rods into an infant or primary grade need not worry any teacher. Provided the minor difficulties mentioned above can be surmounted, the first few lessons pass all too quickly. The attractive colors and precise sizes of the rods stimulate, even in adults, a desire to try them out, to make something, to build, to Play This is exactly as it should be and while the teacher watches the building of houses, trains and fields, patterns, pillars and castles, much is being discovered concerning the basic properties of the rods. Indeed, the aim of the free play lessons and of the directed informal activities shortly to be described, is for pupils to know the rods thoroughly. Before formal study can begin each pupil must develop some facility in manipulating the rods and come somehow to realize

47Â

The Cuisenaire GattegnoChapter Method 3Of Teaching Mathematics Algebra Before Arithmetic Preliminary Exercises

1

that rods of the same color are equal;

2 that the rods have a definite order of size; 3 that they are interchangeable; that, to fit a certain space, such as that taken up by a dark green rod, he can use a red and a pink or two light green or three red rods and so on. It is definitely not the task of the teacher to tell his pupils these things. Indeed, for the first lessons some teachers have the boxes emptied and the rods jumbled on the desk before pupils enter the classroom so that they will not notice that the rods are ordered in size. At the end of the lesson the jumbled rods are scooped back into the boxes without any attempt to pack or check—all this in an effort to let the children discover the above facts for themselves. However, one way or another, either on their own initiative or from situations created by their teacher, pupils must sooner or later discover the inherent properties of the rods before they can effectively proceed to more formal work. In any case, the technique of the first few lessons is simple—the material is simply given out and the children are allowed to play. Perhaps the first thing to be determined is what to call each of the rods. This is best done in personal discussion with the pupils, but where the rods are used in more than one class or grade a prior decision should be made so that the same names are used throughout the school. The following names are commonly used: white, red, light green, pink (some prefer crimson), yellow, dark green, black, tan, blue and orange. These

46 48

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

Â names are not prescriptive, and one is free to make any alterations considered an improvement. Before proceeding further, three possible sources misunderstanding must be removed by assuring that 1

of

the Cuisenaire rods can be introduced at any level;

2 at least some traditional arithmetic is essential at all levels; 3 each stage of the Cuisenaire-Gattegno course, although separated for purposes of discussion, merges into the next in practice. 1

The Cuisenaire rods can be introduced at any level

First, then, while the discussion in this chapter will be definitely related to the kindergarten grade, the Cuisenaire rods may be introduced for use in any grade. It must be emphasized that, whether rods are introduced to become the basis for regular daily lessons in arithmetic or just for one specific purpose, no matter what the purpose or the grade, the procedure should be the sameâ€”the first lessons should be devoted to free play followed by directed informal activities to enable pupils to discover the elementary properties of the rods. Whereas these preliminary steps may take some months in a first grade, most top grades should be ready to proceed after only one or two lessons. It should be realized that while the detailed activities described in this chapter as suitable for kindergarten they can always be adapted for higher grades to ensure the discovery of equivalence in color and length and of order in size.

49 Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

Â 2

Some traditional arithmetic is essential at all levels

A second possible cause of misunderstanding might be the relationship between Cuisenaire and traditional arithmetic in the various grades. It must be stressed immediately that it is essential for both approaches to proceed side by side in all grades. In kindergarten, for example, certain work in ordinal number must be carried out without the rods, while sessions of free play and directed pre-number activities with the rods take place. The development of ability in ordinal number is assured by means of 1

oral counting up to ten to get the sequence of number names;

2

counting to ensure one-to-one correspondence;

3 limiting, that is, developing the ability to select, for example six objects from a group of eight; 4 grouping, that is, developing the ability to construct, for example, two groups of 4; 5 exercises to develop the ability to read, write and recognize the number symbols. In all grades it will be found essential for pupils to understand numeration of numbers beyond those actually being studied with the Cuisenaire material. Thus, during the study of numbers up to 20, numeration and counting up to 100 at least should be understood.

Â

50

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

3 Each successive stage of the course merges naturally into the next The third matter about which there may be some confusion is related to Stage 2 of the Cuisenaire-Gattegno course as mentioned above. Although in actual practice in the classroom there will be no fixed line of demarcation between one stage and the next discussions it will be helpful to distinguish them. Each stage will represent a further development of the childâ€™s mathematical awareness and each stage is dependent for its mastery on full achievement in the stage before it. Consequently, although free play develops some familiarity with the rods and some skill in handling them, the need for directed activity of some kind becomes apparent sooner or later. In this connection, it will be more convenient to subdivide our discussion of Stage 2 (pre-number) into two main headings: 1

directed but informal activities, consisting of games to ensure that the rods are thoroughly known;

2 directed but formal and more intensive activities to ensure that pupils develop certain mathematical concepts essential to an understanding of the arithmetic which is to follow.

Â

51

Â

The Cuisenaire GattegnoChapter Method 3Of Teaching Mathematics Algebra Before Arithmetic Preliminary Exercises

Special Games To Relate Color With Rod Lengths The first group of directed games associated with developing the recognition of the rods by touch as well as sight and hearing may be introduced at the discretion of the teacher at the close of one of the sessions of free play. 1 Pupils are asked to select two or three particular rods, for example, red, pink and light green, and hold them in their hands behind their backs. The teacher (or a pupil) asks them to feel for and produce without looking, say, the light green rod, then the pink, and finally to name the one that is left. This game forces the child to feel for and compare the respective lengths of the rods selected. It can be continued with other sets of rods, the number of rods selected being limited only by the number that can be held comfortably in the hands. 2 Five or six pupils can be made to line up with their hands behind their backs facing the blackboard. The class can see the teacher (or a pupil) placing a rod of a different color in each pupilâ€™s hand. The pupils then turn round and face the class, and each in turn first names and then shows the particular rod given to him. 3 Pupils shut their eyes and feel amongst the jumbled rods for a particular rod named by the teacher (or a pupil). A variation on this particular game would be to cover the jumbled pack with a handkerchief and then ask for a particular rod to be produced.

Â

50 52

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

Directed Informal Activities The next group of organized games has a more mathematical background than those outlined above; at this stage, however, there is no need for, and teachers are earnestly advised against, introducing any mathematical terms whatsoever apart from applying the pupils’ knowledge of ordinal number developed in traditional lessons. The aims of this stage must be kept in mind and the teacher’s task is simply to create situations which develop skill in manipulating the rods and which lead to the observation that 1

rods of the same color are equal;

2 the rods are interchangeable. Many teachers will prefer to give these activities at the beginning of the Cuisenaire lesson before free play is allowed; in any case, whether done first or last, it will be advantageous to continue with free play as part of each lesson for a considerable length of time. The directed activities described below for use in this restricted manner are based on those detailed in Book I, Part II of Mathematics. They are summarized on pages 31-32 of Gattegno’s Teacher’s Introduction. The analysis of them in this chapter will be as follows: 1

making staircases;

2 making mats;

53

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

3 making patterns. The distinction between mats and patterns is purely arbitrary and will be explained shortly. 1

Making staircases

A variety of staircases can be made, the most common of which being the arrangement in order of length, of one rod of each color from white to orange, or from orange to white. This allows for a difference between each step of a white rod (Fig. 26). Staircases with a difference of a red rod, a light green rod or any other rod between each step should also be attempted.

FIG.Â 26.

Â

54

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

It is important to devise a number of activities which involve manipulation of the larger rods. Teachers will be able to make use of other staircases demonstrating the relative lengths of the rods. It is to be remembered that pupils should have experience in building both ways, first from the shortest to the longest length and later from the longest to the shortest. 2

Making mats

This consists in starting from a given length, such as the dark green, and making as many ‘decompositions’ of it as possible using other rods1. The area of the mat should increase with the size of the rod for which the decompositions are made. Further, any ‘mats’ constructed freely by the pupils will vary greatly in detail—rarely will two pupils be found keeping to the same order of decomposition. After each construction, some discussion should take place. For example, after making a mat for the dark green rod, such as in Fig. 27, the teacher (or a pupil) might ask: ‘Who has a red and a pink rod in one line of his mat?’ ‘Who has light green and light green?’ ‘Who has yellow and white? White and yellow? Who has three rods all the same color?’ 1 Fig. 27 illustrates a mat for the dark green rod showing various decompositions of this length.

55

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 27. A ‘mat’ showing some of the decompositions for the dark green rod. After a time, pupils should be given experience in making mats for lengths greater than any single rod. This could be done either with some rod put end to end with the orange rod, or with two or more rods of the same color, for example, two tan, or three yellow, or four orange rods. Included in these activities should be the forming of mats made of rows of one color only. Such a mat for the tan rod, using pink, red and white rods would be an easy one. At a later stage some interesting mats of this kind can be made for the rods which represent such numbers as 12 (orange and red rods end to end), 16 (orange and dark green rods end to end), 24 (two orange rods and a pink rod end to end), 36 (three orange rods and a dark green rod); it will be noted that these numbers have more than two factors.

56

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

FIG. 28. Mat for orange plus dark green using rods of one color only. Such games are more challenging than those first mentioned because they are restrictive. Others can be made more restrictive, for instance when the teacher gives the first rod in each row and asks the pupils to fill up the spaces left, either freely, or with one rod only (or two rods, and so on). It is doubtful whether doing more than a few of these exercises is of much value at the kindergarten level, but they can certainly be used to effect with older pupils. 3

Making patterns

The difference between the terms ‘mats’ and ‘patterns’ is only arbitrary, but in these discussions a ‘mat’ means a variety of decompositions of a given length as in Figs. 27 and 28 above, whereas a ‘pattern’ will mean a single decomposition of a given length (as in Figs. 29, 30 and 31 below). This distinction is made because it occasionally becomes necessary to distinguish between ‘making a mat for the tan rod’ and ‘making a pattern for the tan, using the yellow rod’.

57

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

Patterns Which Provide A Background For Addition Numerous patterns related to addition are possible but they fall into three main types, as follows: 1

Finding a single, rod to fit two or more given rods

A pink rod and a light green rod can be placed end to end, and the pupil asked to find a single rod to fit the length; or red, white and yellow rods may be placed end to end and a single rod found to fit the length, and so on.

FIG. 29. Completing the pattern for pink and light green rods. 2

Finding two or more rods to fit a single rod

Starting with the tan rod, for example, the pupils are asked to find any two rods which together form the same length. Who has used red and dark green? Who has used two pinks? Who has used light green and yellow? . . . and other such questions may be asked.

Â

58

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

FIG. 30

Making a pattern for the tan rod.

3 An extension of the above to patterns involving the use of longer rods The pupils can be asked to make a train, that is, rods placed end to end, of yellow, black and pink rods, and then to find two rods which add up to the same length. ‘Who has two tan rods ? . . . A blue and a black? Make the same train again. Can you find any three other rods that fit? Who has two yellow rods and a dark green rod? . . .’ and so on.

FIG. 31. Pattern involving the use of longer rods.

Patterns Which Provide A Background For Subtraction The purpose of these activities is to compare the lengths of different rods by finding the rod which makes up the difference.

59

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

This, says Gattegno, gives a much truer concept of subtraction than the more generally accepted notion of taking one quantity from a larger quantity. 1

Comparing the lengths of single rods

Take an orange and a tan rod. Place them side by side and look at them. Which is the longer? Which is the shorter? Find the rod which makes up the difference. Try it and make sure it fits the space.

FIG. 32. The red rod makes up the difference between the orange and tan rods. Alternatively, place a yellow rod on top of a blue rod. Which is the longer? Which is the shorter? Which rod is needed with the yellow to make up the same length as the blue? Put the pink rod in the space to see if it fits. Take the yellow rod away. Which rod is left? Put the yellow back on top of the blue. Take the pink from on top of the blue. Which rod is left this time? 2

Exercises involving longer rods

Â

60

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

Put two black rods end to end. Place a yellow rod at the side (or on top) of them. Which is the longer, the two black rods or the yellow? Find the rod which makes up the difference.

FIG. 33.

The blue rod makes up the difference between the yellow and the two black rods.

It is suggested here that the word ‘difference’ be used with kindergarten pupils. This would be in accordance with Gattegno’s assertion that introduction of the correct mathematical terms is advisable from the very beginning. Teachers should realize that pupils can ‘see’ what the term really means—where the word ‘see’ means both ‘to see with the eyes’ and ‘to understand’.

Patterns Which Lead To Multiplication And Division There should be no difficulty in bracketing these operations together. In fact, it will be seen that the particular exercises are also closely related to addition, except that in this case rods of

61

The Cuisenaire Gattegno Method Of Teaching Mathematics

Â

one color only are involved. Examples of four kinds of exercises that can be given to children are outlined below: 1

Find the single rod which fits two or more rods of one color end to end

FIG. 34.

One blue rod fits three light green ones.

2 Find rods of the same color which fit a single rod Take a dark green rod. Which rods of one color placed end to end fit it? Try red rods. How many fit? Try light green rods, rods, white rods.

FIG. 35.

Three red rods fit the dark green.

3 Extension to trains of rods of one color Put two orange rods end to end. Find rods of one color that fit this length when placed end to end. Try pink rods. How many are needed? Now try yellow rods, red rods.

Â

62

Chapter 3 Algebra Before Arithmetic Preliminary Exercises

Or put five tan rods end to end. Which rods of one color fit the length? Try yellow rods. How many are needed? Try orange rods. How many of these are needed ? Or make a train of dark green rods side by side with a train of pink rods. What is the minimum number of rods of each color needed to make the trains equal?

FIG. 36.

A train of dark green rods fitting a train of pink rods.

Of course, these trains could be extended to four dark green and six pink rods and so onâ€”the only limit is the number of rods available. 4 Exercises involving remainders in division Take an orange rod. How many light green rods fit into its length? Which rod fits the space left? How many pink rods fit? Which rod fits the remaining space? Take two orange rods. How many blue rods fit into this length? Which rod fits the space left? How many light green rods fit the tan rod in Fig. 37, and which rod fits into the remaining space?

Â

63

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 37. It will be appreciated that the list of directed informal activities with the Cuisenaire rods is almost inexhaustible. When the examples given above are added to the sessions of free play, it may be found that enough material has been suggested to fill a kindergarten programme of lessons for a full year. Teachers of grades other than kindergarten must adapt the above suggestions to suit the ability, maturity and age of their particular pupils; there seems ample scope for each teacher to alter what has been suggested as he sees fit. The important point, which must be stressed again, is that throughout Stage I (free play) and the informal activities outlined, the spirit of the teaching must be akin to that of playing games. The main purpose is to develop familiarity with the rods and skill in handling them. To achieve this there is generally little need to include all the types of activity mentioned, but just a sufficient amount for the child to proceed confidently to the next stage, the purpose of which will be to formalize the above exercises by the use of whatever mathematical terms the teacher might consider necessary for an effective introduction to arithmetic.

Â

64

Chapter 4 Algebra Before Arithmetic More Intensive Activities

It will have been noticed in Chapter 3 that at no time was there any suggestion that the preliminary activities be graded or presented in any definite order. The reason, of course, is that these activities were to a large extent physical in character. Through a wide variety of experiences pupils were discovering for themselves that the rods had properties on which they could rely, properties which did not vary no matter how the situations changed. It was through this physical activity, through their senses of sight and touch, that pupils were discovering the equivalence relationship between rods of the same color, that rods of different color showed a definite ordered relationship with respect to length, and that these two relationships could be made use of when making different mats and patternsâ€”the algebraic relationship in which two or more rods end to end could make up the same length as another rod.

Â

65

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

Number Or Pre-Number Activities At This Stage? Having developed this far the pupil has the rods at his command for use as tools for the study of whatever mathematical concepts his teacher wishes to present. At this point, two courses are open to the teacher. On the one hand, he can immediately give value to the rods and make them symbols for numbers, choosing the unit to suit his particular purpose. Such a decision would allow him to commence the study of number with his infant class, or to teach some special difficulty in, say, short division to his fourth class. On the other hand, by continuing to use the colors as symbols, he can proceed to demonstrate a large number of mathematical concepts at the algebraic level. This is the course adopted in this book, with the algebra before arithmetic technique devised by Gattegno; it attempts to interpret this theory with as much detail as possible in case the reader decides to adopt it. The aim, with the infant grade, will therefore be to present as many mathematical concepts as are necessary for a firm foundation on which to base later work with number.

Â

66

Chapter 4 Algebra Before Arithmetic More Intensive Activities

Teaching Aims At This Stage A change of emphasis must take place. Whereas previously in the informal activities the pupil was discovering the inherent attributes of the rods, he must now gain insight into certain mathematical concepts. First grade pupils must come to recognize clearly the operations of adding, subtracting, multiplying and dividing, and they should know the meaning of such terms as plus, minus and equals. In order to do so, not only must the activities outlined in Chapter 3 be formalized and their range extended but a much greater degree of mental activity must also be aimed at. Whereas previously one might have been content if the pupils simply observed what they did, saw or touched, now they must also give some expression of what they do, see or feel. It will be agreed that such expression should at least be oral and could, where there are no mechanical difficulties, be written. A much more intensive study of the rod patterns made previously will be essential. The reason for restricting the teacherâ€™s aims during the preliminary activities will now be realized; that is to allow for concentration at this stage on teaching the child 1

to read what his rod patterns say; and

2 to write what his rod patterns say.

Â

67

The Cuisenaire Gattegno Method Of Teaching Mathematics

The child must learn to express whatever basic mathematical concept he creates with the rods, in language and/or by using written symbols.

Factors Associated With The Oral Expression Of Mathematical Concepts With regard to oral expression, the use of proper mathematical language is of the utmost importance. To read what the rods are saying, terms such as plus, minus and equals, and even half, third and fifth should be mastered. Secondly, it will be necessary for the child to have a clear notion of what the rods are saying when arranged in particular patterns. Teachers should realize that the Cuisenaire rods make the meaning of each operation so obvious that this affords little difficulty. Thirdly, the child should understand spoken mathematical statements well enough to interpret them at least by means of rod patterns. Such an understanding should lead to the successful answering of questions such as ‘Which rod is equivalent to the difference between an orange rod and a black one?’

68

Chapter 4 Algebra Before Arithmetic More Intensive Activities

Factors Associated With The Written Expression Of Mathematical Concepts With regard to written expression, there seem to be three main factors: 1

writing the names of the colors;

2 writing the mathematical signs for the various relationships; 3 combining these two abilities to express any given rod pattern in a correctly written mathematical statement. This last, of course, would involve not only the ability to write simple statements from the rods but also the reverse ability of interpreting written or oral statements in terms of rod patterns. With mastery in both these directions the completion of any mathematical statement either by using the rods, or orally, or in writing, becomes a means whereby a wide variety of exercises can stimulate the mental power of pupils. These then are the tasks to be undertaken by the teacher during this part of the course. The only hindrance to the carrying out of such a programme would be the lack of technical skill in writing. Where the writing skills have been adequately developed, the full programme can be covered. But to apply Gattegnoâ€™s theory objectively and in its entirety to a first grade, where the writing of letters and

Â

69

The Cuisenaire Gattegno Method Of Teaching Mathematics

Â

numbers is just being mastered, is a serious challenge. To delay it could mean delaying the introduction of Cuisenaire work in number until the beginning of the second grade. Even if number were introduced at the beginning of the third term, the standard reached would in many ways be equivalent to that reached at present by pupils at the end of their second grade. This seems to be where Gattegnoâ€™s theory is so revolutionary. An interesting experiment which could be attempted (but not before the teachers concerned have had some experience with this method) would be to compare the progress of a first grade such as the one described above with another one also using the Cuisenaire-Gattegno method, but to which number work would be introduced immediately after the oral activities with names-names are covered. The latter class would therefore experience written expression only in terms of number.

The Place Of Oral Discussion In The Cuisenaire-Gattegno Method Although the reader may have detected in the above sections the close bond which exists between oral and written work, he may also agree that this second stage of the method could be subdivided into three fields of activity as follows: The Cuisinaire Gattegno Method Of Teaching Mathematics

1

Â

manipulating the rods and observing (described in Chapter 3);

70

Chapter 4 Algebra Before Arithmetic More Intensive Activities

2 manipulating the rods and reading what is seen in the patterns; 3 manipulating the rods and writing what is seen in the patterns. In practice it will be found that there is no clearly defined division between these stages. However, it will also be found that oral discussion provides the key to successful written work. Reading the rod patterns in mathematical language is the guiding force which leads to an understanding of the writing, and this is so at all stages of the Cuisenaire course. For this reason, teachers should never be in a hurry to introduce written work.

Name-Name Notation During the informal activities described in Chapter 3, it is possible to show pupils that the initial letters of the names of the colors can be used as a â€˜shorthandâ€™ notation. In this way the written symbols for the rods could be introduced gradually and games devised (similar to those played to relate color and length) to ensure that the relation between each rod and its written symbol was known. Additional activities for this purpose are suggested as follows: 1

Â

making a staircase and writing down the symbol for each rod in order from bottom to top;

71

The Cuisenaire Gattegno Method Of Teaching Mathematics

2 taking up a handful of rods; looking at those with different colors; writing the symbol for the longest, the shortest; 3 making a train with a number of rods; writing down the symbol for each rod in the train; 4 making a mat for the yellow rod; writing down the symbols for the rods in the pattern. However, there are difficulties encountered in making a decision concerning the written symbols for certain rods. There is little difficulty in finding a symbol for seven of the rods; the letter g is used for the light green rod and should cause no confusion with d for dark green. w—white r—red g—light green p—pink y—yellow d—dark green o—orange The three b’s for black, brown and blue, on the other hand, present a problem. Some prefer the following convention: black—b brown—B blue—bu

72

Chapter 4 Algebra Before Arithmetic More Intensive Activities

Gattegno, however, suggests that the brown be called ‘tan’ and that we write black—b tan—t blue—B This notation has been used throughout this book; teachers may wish to adopt some other scheme, but no matter what names are used it is important that pupils in the various grades of a school using the rods all adopt the same names.

Gattegno’s Exercises In Qualitative Arithmetic In Part II of Book I of Gattegno’s Mathematics a wide variety of exercises are suggested which form a basis for teaching mathematical relationships. While no mention is made of formal mathematical expression either orally or in writing it must be surmised from Gattegno’s publications that, at the teacher’s discretion, full use can be made of the exercises for either or both these purposes. The exercises form the basis upon which any teacher experienced in the Cuisenaire-Gattegno method can build an infinite variety of activities. Moreover, most of the exercises, if not all, are simply suggestions and can be attempted for any one of three reasons:

73

The Cuisenaire Gattegno Method Of Teaching Mathematics

1

as an informal activity aiming simply to secure dexterity in handling the rods;

2 as a formal activity aiming at correct oral expression; 3 as a formal activity aiming at correct written expression. Consider, for example, exercise 39 which reads: ‘Make trains that are all tan and orange. Can they be the same length?’ The following analysis of the exercises, with their numbers in parentheses, will show the progression as recommended by Gattegno and will also form the basis of a programme of lessons for this particular stage of the course. 1

Exercises chiefly involving addition (nos. 6-14 and 25-28) 1

Mats (6-14) 1

Making mats independently (2)

2 Making mats for specific rods (3, 4) 2 Patterns (14-17) 1

Single rod—finding the rods which fit into the particular length

2 Two or more rods—finding the single rod which fits into the particular length With these exercises as a basis, the terms ‘plus’ and ‘equals’ can be introduced, first orally and later in writing.

74

Chapter 4 Algebra Before Arithmetic More Intensive Activities

2

Exercises chiefly involving subtraction (nos. 5-13 and 18-19) 1

Finding rods smaller than, bigger than, and equal to a given rod (5-7)

2 Comparing the lengths of pairs of rods (8-9) 3 Staircases of rods with common differences (10-12) 4 Exercises with specific rods (13) 5 A variation in procedure (18-19) The terms ‘minus’ and ‘difference’ can be introduced at this stage. 3 Exercises chiefly involving multiplication and division (nos. 20-30) 1

Finding a single rod which fits a train of rods of one color only

2 Finding a number of rods of a particular color which fit a single rod 3 Remainders 4 Comparing trains of rods of one color only The term ‘times’ can be introduced here. 4 A variety of exercises establishing the notions of equality and of wholes and parts (nos. 31-34).

75

The Cuisenaire Gattegno Method Of Teaching Mathematics

Exercises Involving Addition Of the exercises involving addition, nos. 1 and 2 are free activities related to those mentioned in Chapter 3 and aim simply at observation. It can be noticed, however, that in nos. 3 and 4 there is a hint of systematic progression in the making of mats from the light green to the orange. Fig. 38 below shows a mat for the dark green rod from which some examples will be taken to illustrate this part of the discussion.

FIG. 38.

The Meaning Of ‘Plus’ As the present aim is the correct expression of mathematical concepts, the terms ‘plus’ and ‘equivalent’ should be used even

76

Chapter 4 Algebra Before Arithmetic More Intensive Activities

by first grade pupils. Gattegno argues that a child’s name, such as ‘Marjorie’ is no more difficult to understand than a mathematical term like ‘plus’. There seems no doubt that with the assistance of the Cuisenaire rods and after a few similar lessons for consolidation, the meaning of the term ‘plus’, together with the meaning of the operation of addition can be seen and understood.

The Meaning Of ‘Equivalent’ Should one be content with the use of the word ‘makes’ (for example, pink plus red makes dark green), or with ‘are the same (length) as’ instead of insisting on ‘equals’ or ‘is equivalent to’? Gattegno, of course, would say ‘No. Commence with the correct mathematical term’. In the psychologically correct Cuisenaire lesson, attention can be turned to the rods to show equality in any of the many ways that are possible. For example, pupils can be instructed as follows: 1 Find any two rods that together equal the dark green rod. Put them end to end and see if together they equal the length of the dark green. Now read what the rods are saying; or 2 Find two rods that together equal a train of any three rods. Read what your rod pattern is saying.

77

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 39. The child reads from his pattern: ‘Pink plus light green plus red equals dark green plus light green’. Once again the rods, much more than any other concrete medium, give a very clear notion of equality of length or equivalence of patterns.

Introducing The Written Symbols For ‘Plus’ And ‘Equals’ It is only after such terms can be used confidently in oral discussion and after pupils have mastered the idea of using initials as symbols for the colors that the written signs should be introduced. The steps taken by pupils towards mastery of the written equation are set out clearly below: First step: Making the mat for the yellow rod.

78

Chapter 4 Algebra Before Arithmetic More Intensive Activities

FIG 40. Second step: when the color symbols are known the mat could be written this way. $$Image$$ y p g r

r

w r w

Third step: when the written symbol for ‘plus’ is known the mat could be written this way. p+w g+r r+r+w Finally, when the written symbol for ‘equals’ is known, the mat could be written in correct mathematical notation. p+w=y

79

The Cuisenaire Gattegno Method Of Teaching Mathematics

g+r=y r+r+w=y or y=p+w y=g+r y=r+r+w A further important principle is that the child should discover the need for each sign himself. The search for a suitable written sign for a particular situation could become an interesting and profitable lesson.

The Meaning Of ‘Minus’ The comparative ease with which the mathematical terms relating to addition can be introduced for the purposes of both oral and written expression is illustrated again for those related to subtraction and multiplication. The pupils read from their rods that ‘the blue rod minus the yellow rod equals the pink rod’ or that ‘the tan rod equals four times the red rod’. Once the word is known it is an easy step to express the operation in writing

80

Chapter 4 Algebra Before Arithmetic More Intensive Activities

FIG. 41.

b–p=g b–g=p

Reading And Writing The Symbol For Multiplication The same process can be carried out for multiplication. For example, the pupil can be instructed in the following manner: ‘Take a red rod; take another; and another. How many times have you taken a red rod?’ This leads in turn to the expression that ‘three times the red rod is equivalent to the dark green rod’, which is finally written

81

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 42.

3 × r = d or d = 3 × r

The need to extend the notion of equality from the addition situation to that of multiplication and subtraction is obvious.

Division Gattegno, in his Mathematics, does not use the word ‘divide’ or the division sign until towards the end of the study of numbers up to 20 in Book I. He includes an abundance of exercises, however, some set out in a form similar to the following: ‘How many red rods can we put end to end to equal the length of an orange rod?’ Other exercises involving remainders are of the form: ‘How many light green rods fit into the length of the tan? Which rod would fill the remaining space?’ Exercises such as these must be included as part of the experiences of first grade pupils.

82

Chapter 4 Algebra Before Arithmetic More Intensive Activities

Fractions Fraction relationships receive unusual emphasis in the Cuisenaire-Gattegno approach. Various concepts of a fraction can be demonstrated to, and experienced by, pupils just as simply as any of the other basic operations. Once children know the terms ‘half, ‘third’, fifth’, and so on, they can easily express the fact that the red rod is half the pink, a third of the dark green and a fifth of the orange. Exactly the same progression from oral to written activities is possible here as it was for addition, subtraction and multiplication.

It is the purpose of this stage to allow first grade pupils to discover and understand the reading and writing of such fractions as a half, a third, two thirds and three tenths. This is certainly revolutionary. It is stressed that exercises such as the following are within the scope of children who are able to count up to ten.

83

The Cuisenaire Gattegno Method Of Teaching Mathematics

1

Make a train with three red rods. Take another red rod. This one is called one third—written —of the length of the train.

2 Make a train of 3 yellow rods. Find

of it.

Reverse Processes So far, then, pupils have mastered the formal expression, both orally and in writing, of the operations of addition, subtraction, multiplication and, to a certain extent, division. They can read or write the mathematical statements involved in any mat or pattern through their understanding of such terms as ‘plus’, ‘minus’, ‘times’ and ‘equals’. Although exercises in reverse processes will not be emphasized till a later stage, it should be possible for first grade pupils to make at least some of the statements shown with the following patterns.

y=g+r y=r+g g+r=y r+g=y

y–g=r y–r=g r=y–g g=y–r

84

Chapter 4 Algebra Before Arithmetic More Intensive Activities

FIG. 44.

Addition and subtraction relationships.

B=3×g

×B=g

3×g=B

g=

g+g+g=B

B–g–g–g=0

B=g+g+g

How many light green rods fit into the length of a blue rod?

FIG. 45.

×B

Multiplication and division relationships.

Exercises In Mixed Operations Another possibility concerns the variety of exercises involving more than one operation or ‘mixed operations’ as they are commonly called. For example, the patterns in Fig. 46 and Fig. 47 illustrate the ease with which young children can understand the accompanying statements.

85

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 46.

FIG. 47.

o = (2 × g) + p o = p + (2 × g) p = o (2 × g) o—p=2×g

o + g = p + y + (2 × r) p + y = (o + g) – (2 × r) p + (2 × r) = (o + g) – y g + o – (y + r) = r + p

Completing Patterns In Writing It will have been noticed that this discussion has been concerned with enabling pupils to read or write a mathematical statement from the rod pattern. This, of course, is fundamental and must be accomplished before the pupils can successfully proceed to such fields as 1

making a rod pattern from a mathematical statement given either orally or in writing;

2 completing rod patterns; 3 completing patterns in writing.

86

Chapter 4 Algebra Before Arithmetic More Intensive Activities

It should be noted that Gattegno’s expression, ‘Complete these patterns in writing’ seems to indicate that only the answer is required and not the complete statement. The full statement is, however, required when expressing a rod pattern orally or in writing. The pupils can be asked to write from memory five sums about the blue rod, for example: B=3×g g= ×B

r+b=B B – (2 × r) = y

w+t=B

The Part Played By Visual Images Finally, a word must be said about exercises such as Nos. 18, 35 and 36 of Gattegno’s Mathematics, Part II, Book I. This type of exercise is common throughout the course and helps considerably in the development of visual images, which are the first step towards abstraction.

87

Chapter 5 The Study Of Numbers Up To 10

Application Of Gattegnoâ€™s Algebra Before Arithmetic Theory Irrespective of the purpose the rods are intended to serve and of the grade for whom they are designed, at least some sessions of free play and directed activity are essential to enable pupils to handle the rods efficiently and to let them discover some of their properties. Once this is done, the rods can be put to any one of a large number of uses, according to the particular needs of the class. An indication was given in Chapter 4 of the extent to which the Cuisenaire rods can be used for directed activities, made distinct from the informal activities described in Chapter 3 by their particular aim of introducing pupils to certain mathematical terms and concepts fundamental to the understanding of

89

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

arithmetic. By using what is perhaps the most attractive feature of the rods, their color, it was found that pupils in a first grade could confidently and correctly express many quite complicated concepts both orally and in writing. The semi-abstract qualities of the Cuisenaire rods allow them to play a dual role in the teaching of arithmetic in particular and mathematics in general. They provide clear and concrete experiences of number behavior as well as insight into algebraic relationships. We have been concerned up till now with the nature of the four operations. The aim at this stage will be to broaden the childâ€™s experience from the study of mathematical relationships in terms of names-names to the application of the same relationships to numbers. Just how this is done will be considered after the problem of actually giving numerical value to the rods has been discussed.

Giving Numerical Value To The Cuisenaire Rods It must be emphasized from the very beginning that for the purpose of ascribing number-names to the rods any rod can be chosen as the unit. Although a large proportion of the basic work in number is carried out with the white rod as the unit, there are many occasions, for instance in connection with fractions or with the representational use of the rods in problems, when it is essential for pupils to know that the rods are not numbers but mere pieces of wood which can be used to represent numbers. The child must realize as soon as possible that the yellow rod,

Â

90

Chapter 5 The Study Of Numbers Up To 10

for example, is no more ‘five’ than the figure ‘5’ written on the blackboard, or five pigs in a sty. Numbers are abstract concepts which can be symbolized in many different ways, sometimes by figures like 5, sometimes by words like ‘five’, and so on. The Cuisenaire rods themselves help the child considerably towards this realization. He is reminded incidentally and frequently of this fact, for example when he decomposes the yellow rod into a red and a light green or into a white and a pink. Two plus three and one plus four are also five, and he gradually discovers that many other patterns can also symbolize the number five, for instance, nine minus four. With this important fact in mind, the following method is perhaps the best way of introducing numerical value to the rods. If a particular group of children are ready for number work it is a simple matter to hold up the white rod and ask: Which rod is this? (The white one). How many white rods are there in my hand? (One). If the white rod is one, what is the name of this ? (Holding up the red. Answer: two). Why? (Because two white rods fit the red; or, measured with the white one the red is two). In this or similar manner and using the white rod as the unit it is a simple matter to determine the values of all the rods. Having done this, the child should immediately be asked:

91

The Cuisenaire Gattegno Method Of Teaching Mathematics

What color is this rod? (Red). How many rods are there in my hand? (One). Well, if this rod is taken as the unit, what is this rod called? (Holding up the pink. Answer: two). Why? This can be repeated using any of the rods as the unit. In this manner it will clearly be established that the rods are objects, that they have no set numerical value in themselves, and that the values accorded to them depend on the units of measurement. During the rest of the course, it is important to return to this idea quite frequently, by holding up the orange rod, say, and asking: What is this? The answer should be ‘an orange rod’, but if the answer given is ‘ten’, one should ask: How many rods are there in my hand? (One). What is the color of this rod? (Orange). Well then, what do you actually see? (One orange rod).

92

Chapter 5 The Study Of Numbers Up To 10

Games To Relate Color, Length And Numerical Value The idea that each rod can have value only in accordance with its relationship to other rods having been introduced, the next step will be to ensure that the pupils can confidently relate the rods to their respective values with the white as the unit, as most of the work in arithmetic will require that the rods have these values. First, there could be a repetition of the games played earlier, in which the object was for pupils to learn the relation between color and length. An additional exercise might be to ask pupils to say how many white rods fit into the lengths of other rods which are shown to them one at a time.

Color Names For Reference Purposes When these games are being played, and as a general rule throughout the course, the rods should be called by their color names, not their number values except, of course, when a numerical answer is required. For example, for the game mentioned above, the following questions give the correct terminology. Take a black, a pink and a dark green rod; (not ‘a 7, 4 and a 6’);

93

The Cuisenaire Gattegno Method Of Teaching Mathematics

How many white rods fit into the black? Into the pink? the dark green? the pink plus dark green?

Staircases To Relate Color, Length And Numerical Value A second series of exercises leading to recognition of numerical values would be by means of staircases. For instance, for the staircase with a common difference of a white rod, the game could consist of touching each rod in turn whilst naming its value and moving either up or down the staircase. Variations on this game could be as follows: 1

the teacher (or a pupil) names a particular rod for the pupils to find and name;

2 pupils write the value of the rods in order, moving either up or down; 3 the teacher names rod colors at random and the pupils write down the corresponding numerical values; 4 pupils shut their eyes and ‘read’ the staircase, going up or down;

94

Chapter 5 The Study Of Numbers Up To 10

5 a pupil removes one rod from his neighborâ€™s staircase and asks him to give the value of the rod removed. Such exercises would also apply to staircases with common differences other than the white rod. A wide variety of these is possible. For example, three staircases can be made with the light green as the difference between successive steps:

FIG. 48. FIG. 49.

Light green, dark green, blue (3,6,9). White, pink, black, orange (1, 4, 7, 10).

FIG. 50.

Â

Red, yellow, tan (2, 5, 8).

93 95

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

Systematic Grading Of Number Is Not Essential The first step to be taken after the recognition of the number values ascribed to the rods is to require pupils to read their rod patterns using number names instead of the color names used previously. In following this procedure, it is immaterial which particular numbers are used and there is little need to keep to any systematic numerical order as it is the development of oral expression which is being aimed at. It will now be realized that the more completely Gattegnoâ€™s theory concerning the introduction of relationships through rod patterns has been followed, the simpler the present task will be. However, where pupils have attained ability only in the oral expression of these relationships and have not attempted any written transcriptions of them, the procedure might be as follows: The Cuisinaire Gattegno Method Of Teaching Mathematics

1 First oral expression of relationships in terms of number might be developed by spending some time effecting a transfer to the reading of rod patterns in terms of numbers up to ten;

Â

96

Chapter 5 The Study Of Numbers Up To 10

2 Then, using this newly-acquired ability in oral expression, one might proceed to the written expression of rod patterns, dealing first with numbers up to 5 and then with each of the numbers 6, 7, 8, 9 and 10 in turn.

Steps To Develop An Understanding Of Equality 1

Study of basic equality (e.g. 10 = 10) showing that, no matter how placed, equal rods remain equal.

2 Two rods end to end can be equal to another rod; e.g. 3 + 7 = 10. 3 One rod can be equal to a train of two or more rods, e.g. 10 = 3 + 2 + 5. 4 Two rods can be equal to two other rods; e.g. 3 + 7 = 4 + 6. 5 Extending this to longer trains: 2 rods = 3 rods; 4 rods = 5 rods; e.g. 4 + 6 = 3 + 2 + 5. 6 Showing that if equal rods are added to equal rods the results are equal; e.g. 4 = 4 and 6 = 6, therefore 4 + 6 = 4 + 6. 7 Showing that if equal lengths are subtracted from equal lengths, the results are equal; e.g. 10 = 10 and 7 = 7, therefore 10 – 7 = 10 – 7. 8 One rod can be equal to a train of rods of one color; e.g. 10 = 5 × 2.

97

The Cuisenaire Gattegno Method Of Teaching Mathematics

9 Trains of rods of one color can be equal to trains of rods of another color; e.g. 5 × 2 = 2 × 5. In all these cases, it must be emphasized that the reverse equalities would also be true, e.g. 3 + 7 = 10 and 10 = 3 + 7. One of the ways in which awareness of equality could be developed is as follows: Find two rods that equal the orange rod (named by its color not its number name). Read your pattern out loud using numerals (‘Four plus six equals ten’). If the pupils are able to write down what they see, they could put down 4 + 6 = 10 or 10 = 4 + 6; or 6 + 4 = 10 or 10 = 6 + 4. The patterns for steps (iv) and (v) above could be read (or written) in a variety of ways.

FIG. 51

98

Chapter 5 The Study Of Numbers Up To 10

The above pattern could be rearranged for different readings (or writings), which would yield

and or again and and so on.

4 + 6 = 3 + 2 + 5 or the reverse 3 + 2 + 5 = 4 + 6 6 + 4 = 3 + 2 + 5 or the reverse 3 + 2 + 5 = 6 + 4 4 + 6 = 2 + 5 + 3 or the reverse 2 + 5 + 3 = 4 + 6 6 + 4 = 2 + 5 + 3 or the reverse 2 + 5 + 3 = 6 + 4

Understanding Addition This understanding has in large part been achieved at an earlier stage, when decompositions (or partitions) of a rod into smaller rods which together add up to its length were practiced. These exercises can be repeated here, using the numerical values of the rods for both oral and written work. The important property of addition, that of commutativity, can also be emphasized at this stage. The rods can be used to advantage to demonstrate relationships such as 2+3+5=2+5+3=3+5+2=3+2+5=5+2+3=5+ 3 + 2 = 10

Â

99

The Cuisenaire Gattegno Method Of Teaching Mathematics

Understanding Multiplication Multiplication, being in reality repeated addition, can be introduced as a development of the study of addition by asking the children questions of the following type: Can you make trains of rods of one color only which add up to the length of the orange rod? Which rods have you used in your trains? How many yellow ones are there? How many red ones? How many white ones? Is it true that two times five equals ten? Do you agree with the following writings: 2 + 5 = 10, 5 + 2 = 10, 10 × 1 = 10? The terms ‘factor’ and ‘product’ can also be introduced at this stage.

Understanding Subtraction 1

Questions of the following type can be asked:

100

Chapter 5 The Study Of Numbers Up To 10

What must be added to the black rod (or to 7) to make a train equal to the orange rod (or to 10)? What must be added to the light green rod (or to 3) to make a train equal to the orange rod? What must be added to two red rods (or to 2 × 2) to make a train equal to the orange rod? What must be added to a train made of one red and one pink rod (or to 2 + 4) to make it equal to the orange rod? 2 A vocabulary change is made and the above questions repeated in the following way: What is the ‘difference’ between the orange and the black rods? Or between the orange rod and a train made of a red and a pink rod? 3 Instead of ‘difference’ the term ‘minus’ is now used: Which length is equal to ‘the length of the orange rod minus the length of the black rod’ or, in short, to ‘the orange minus the black’?

Understanding Division 1

The children are asked: How many yellow rods equal one orange rod? And how many red rods?

101

The Cuisenaire Gattegno Method Of Teaching Mathematics

2 Then they are asked: How many light green rods equal one orange rod? Is there a gap left? Which rod do you need to fill it? This is repeated with other rods; the remainders are studied. 3 The following questions can now be asked: Which length is equal to orange minus light green (10 — 3)? orange minus two light greens (10 — 3 — 3 or 10 — (2 × 3))? orange minus three light greens (10 — 3 — 3 — 3 = 10 — (3 × 3))? This stresses the nature of quotition division. See Chapter 6. 4 The children can also be asked: Pick up an orange rod. Can you find two equal rods which together are equivalent to it? Which are they? This lays the basis for study of partition division. See Chapter 6. Gattegno, in his Mathematics Part IV, Book I arranges his exercises so that equality and the basic operations are studied first on numbers up to 5 and then, seriatim, he takes the numbers 6, 7, 8, 9 and 10.

102

Chapter 5 The Study Of Numbers Up To 10

The Four Operations With Numbers Up To 5 The first exercise is concerned with measuring the ten rods with the white one as the unit and introducing the relevant numerical notation. This is followed by exercises involving the formation and writing down of mats. In exercises 2 – 10, addition and subtraction are introduced almost simultaneously. In exercise 2, the sign ‘plus’ is used. In the next exercise the ‘minus’ sign is introduced; rod patterns for both addition and subtraction are to be read in correct mathematical language, and the correct reading of mathematical symbols (3 – 2 = 1 and 2 – 1 = 1) is required. Exercise 4 includes problems (patterns to be completed in writing) involving almost all the possible variations on the addition and subtraction of the numbers 1, 2 and 3. Both reading and writing of mats involving numbers up to 5 are required in exercise 8; pupils read the written symbols in both 8 and 9, and complete patterns in writing in 7, 8 and 10. The brief outline of these few exercises will indicate the importance of understanding the background and general principles of the Cuisenaire-Gattegno method before attempting the classroom use of the Mathematics series of books. Multiplication in connection with numbers up to 5 is the subject of exercises 11-18, which also include a variety of problems in mixed operations. In exercises 19-41, both types of division are considered, that is,

103

The Cuisenaire Gattegno Method Of Teaching Mathematics

1

Quotition (for example, how many red rods make the length of a pink rod?)

2 Partition (for example, what is half of 4?) A more detailed analysis of this group of exercises (1-41) would show that Gattegno expects mastery of oral and written expression of the four operations in all their varied forms, up to and including the conventional vertical notation.

The Use Of Brackets There is, however, a further accomplishment which needs special discussion. It is doubtful if simple brackets such as those first appearing in exercise 10: e.g. 5 – (2 + 1), will have been used by primary pupils, this being left for introduction in secondary school. Yet now, by means of the Cuisenaire rods, the use of brackets can be understood by pupils in our infant grades. Brackets are essential for the correct expression of certain subtractions, and if from the very beginning every situation is exploited to the full, they should be introduced when the child commences written work. For the purpose of our discussion, numbers will be used in this explanation, although brackets can be introduced using the names-names of the rods. Once again the first step is oral discussion of a rod pattern. Fig. 52.

104

Chapter 5 The Study Of Numbers Up To 10

As addition the pattern in Fig. 52 reads: 8 = 3 + 2 + 3. It can be read in a form which is equivalent to ‘eight minus three plus two equals three’, where the utterance of the words ‘three plus two’ is speeded up. As the child hears that the ‘three plus two’ is taken together he would want to give it a distinctive written form. When this is first written, the child could write 8 – 3 + 2 = 3 and then show that 8 – 3 = 5, and 5 + 2 = 7, thus producing an answer quite different from the one expected. If the earlier discussion has been adequate, the child will explain that he meant 8 minus 3 plus 2 together. One of two approaches can now be taken: 1

He can be given the written symbol which means ‘3 plus 2 together’, and its name ‘brackets’,

or 2

when the child reads his pattern, he can be asked to cup his hands on either side of the rods he means to keep together, and then, when the fact is to be written, he simply draws his hands.

FIG. 53.

8 – (3 + 2) = 3.

A third step would be to cultivate the notion that the contents of the brackets can be reduced to a single term, when the brackets

105

The Cuisenaire Gattegno Method Of Teaching Mathematics

will no longer be necessary; that is 8 – (3 + 2) = 3 becomes 8 – 5 = 3. FIG. 54. A game which could prove useful at this stage would be to start with the statement ‘eight minus something equals three’. The pupils are asked to fill the space, each child providing his own contribution. For example: 8–5=3 8 – (3 + 2) = 3 8 – (4 + 1) = 3 8 – (2 × 2 + 1) = 3 8 – (5 × 1) = 3 8 – (2 + 3) = 3 etc.

Exploring The Number System While pupils are developing the use of brackets and when they have achieved some skill in written expression, they should be given the opportunity to explore the number system each according to his individual talents. Even if their knowledge of numeration has not gone beyond the range of numbers set for study with the rods, teachers will be surprised and sometimes amazed at what the children can discover for themselves.

106

Chapter 5 The Study Of Numbers Up To 10

The following is an example of a game which can be played with the whole class. It aims at giving children greater flexibility in their outlook on number relationships. Make the pattern red plus pink plus light green equals blue. Read your pattern. (2 + 4 + 3 = 9) Let us think of other ways of expressing this relationship. What can we replace 2 with? 2 = 10 – 8;

of

of 8; 19 – 17;

of 9; . . .

What can we replace 4 with? 4=

of 8;

of 5; 100 – 96; 2 × 2; . . .

What can we replace 3 with? What can we replace 9 with? (For each number a wealth of equivalents is provided by the pupils.) Now let us see what we can replace 2 + 4 + 3 = 9 with. of 9 + (100 – 96) + or

of

of 8 +

or again (10 – 8) +

of 6 = 4 + 5

of 5 + (20 – 17) = 2 × 4 + 1

of 16 + 3 × 1 = 107

of 18

The Cuisenaire Gattegno Method Of Teaching Mathematics

Can we use one of the other ways of writing the pattern and find equivalent expressions for each of the numbers? Let us use 9 – 2 = 4 + 3; of 10 – (19 – 17) = 2 × 2 +

of 6

Now repeat this with a pattern of your own choice. This is where achievements become more and more surprising. Pupils gradually discover their power to manipulate larger and larger numbers and thoroughly enjoy writing simple relationships in the most complicated ways. A seven year old second grade pupil, for example, wrote these sums from his mat for the number 13: of 39 = 2 × 2 + 3 × 13 × 1 = of 68 +

of 100 + of 2 =

of 6 +

of 6

of 20 + of 56 +

of 1000 + 1 of 50

Progression In Written Exercises In Gattegno’s exercises extensive knowledge of notation is expected from the pupil with no attempt at the usual gradation according to difficulty. This lack of gradation is rendered possible by the ability of the child to manipulate his rods as required by each particular exercise.

108

Chapter 5 The Study Of Numbers Up To 10

1

From the rods to a written symbol

The child should at this stage be capable of reading and writing what the rods say once a pattern has been made. Recording what he has just discovered with his rods should present no difficulty if he has the necessary knowledge of the written conventions at his disposal. 2

From written symbols to the rods

The reverse process of interpreting a written statement in terms of a rod pattern has already received some attention during oral activities. When told to make the rod pattern showing that three plus four equals five plus two, the child usually has no difficulty. However, when the same statement is written on the blackboard and the child simply sees 3 + 4 = 2 + 5, there is sometimes some hesitancy in translating it into a rod pattern. Thus, a second step towards mastery of written exercises is to require the making of rod patterns from a wide selection of written statements. 3

From written symbols direct to an answer

At this stage, the child is presented with an incomplete expression, e.g. 3 + 4 = 2 + , which has to be completed in writing. If unable to complete it he must resort to his rods, but after a while he will write the answer down directly without recourse to the rods at all. The question is frequently asked: ‘But when are the rods discarded altogether?’ The above provides the answer—the child

109

The Cuisenaire Gattegno Method Of Teaching Mathematics

Â

discards his rods naturally and automatically whenever he understands the situation involved. It is always a private and individual decision made by each child for himself with never a hint of pressure from his teacher. This progression in written work may be summarized thus: 1

the child works from the rods to a written symbol;

2 he works from written symbols to the rods; 3 he works from written symbols to the rods to an answer; 4 he works from written symbols direct to an answer.

Varying The Exercises A natural extension to looking at an incomplete pattern and mentally estimating the space is to hear only the names of the rods (either color names or number names) and say which rod is needed. A definite direction is given for pupils to shut their eyes, to think of the rods concerned and then to say which rod is needed. This is not quite the same as completing patterns by going direct to the answer (mentioned above), or even the same as writing sums from memory (such as is sometimes attempted). Such exercises involve an effort, directly stimulated by the teacher, to recall visually a particular pattern.

110

Chapter 5 The Study Of Numbers Up To 10

These exercises encouraging visual images as steps towards mental arithmetic, are summarized in exercise 18 of Part II, Book I of Mathematics. This suggests that any exercise can be done 1

by using the rods (normal routine);

2 by looking only, and showing the rod that is needed; or 3

by hearing only the names of the rods and saying which rod is needed.

The main purpose of this chapter has been to effect the transfer from the study of rod relationships in terms of names-names to the study of arithmetic in terms of number-names up to 10 and the oral and written expression of these number relationships. This study of arithmetic will of course be intensified when large numbers are tackled.

109 111

Chapter 6 Teaching The Basic Concepts Of A Fraction

Quotition And Partition In Division One important contribution made by the Cuisenaire-Gattegno method in the teaching of arithmetic is the clear distinction drawn between quotition and partition division. There has always been some confusion in the meaning of the division sign used for both these operations, particularly in work with such problems as those discussed below. 1

How many six penny ice creams can I buy for 2/–?

This is customarily, though incorrectly, written 2/ – ÷ 6d. = . It involves finding a quotient, finding the number of equal groups valued at 6d. contained in 2/–. The size of the equal groups is

113

The Cuisenaire Gattegno Method Of Teaching Mathematics

known and the problem is to find how many there are; 2/– is equivalent to 24 pence and contains four equal groups of 6d. and therefore the answer is four ice creams. This is an example of quotition (finding a quotient) and corresponds to such questions in arithmetic as ‘How many fives in fifteen?’ 2

Share 2/– equally between six boys.

This, on the other hand, involves dividing 2/– into six equal parts, each of which would be called one sixth. The size of the equal parts is not known. The problem should be set out as 2/– and therefore involves the operation of partition, separating a quantity into a given number of equal parts in order to find the size of the parts. However, as pupils in lower primary grades do not clearly understand the meaning of one sixth, one is generally satisfied with the same setting out as in the operation of quotition, that is 2/– ÷ 6 = . The answer in this case is, of course, fourpence; a similar example in arithmetic would be: ‘Share fifteen objects equally between five boys.’ The lack of understanding of such concepts as seems to be the basic reason for the confusion in the use of the division sign for the operation in both quotition and partition.

114

Chapter 6 Teaching The Basic Concepts Of A Fraction

Quotition The example for quotition ‘How many fives in fifteen?’ would be written 15 ÷ 5 = ; this the child should read as ‘fifteen divided by five’. He would use the yellow rod to measure the length representing 15 to find his answer. Gattegno appreciates the difficulty of associating the division sign with ‘How many . . . ’ , and therefore uses numerous examples of quotition throughout the study of operations involving numbers up to 10 and 20, all in the form of ‘How many . . . in . . . ?’ He does not introduce the sign for division until comparatively late in the course. (See exercise 41, Part IV and exercise 48, Part VI of Book I of Mathematics.) His purpose is to let pupils mature with the operation of quotition before introducing the written sign. He goes on to say: ‘It is usually assumed that the inverse operation of multiplication is division. In a sense this is so, but the word is ambiguous and is loosely used to refer to the procedure by which quotient and remainder are found’ (Numbers in Color, p. 8). To perform such a division, he says, one should ‘take one rod and use it over and over again’ to measure how many ‘go into’ the given number. In the opposite sense, division is basically repeated subtraction and 15 ÷ 5 means 15 – 5 – 5 – 5 where the fives are counted successively as one subtracts. Such a notion of division provides the meaning of

115

The Cuisenaire Gattegno Method Of Teaching Mathematics

‘remainder’ because in the example 19 ÷ 5, after the three fives are subtracted, four is found to be the remainder.

Partition Partition is best described as sharing, and in order to share fifteen between five children one must be able to find one fifth of fifteen. By making five light green rods fit the length representing 15, the child would discover that × 15 = 3. If the relationship of 3 to 15 is one fifth, then the inverse relationship, that of 15 to 3, is the multiplication five times three. Gattegno continues (on p. 8 of Numbers in Color): ‘When the operation of multiplication is reversed, fractions are the outcome’ (not division); and: ‘When fractions are taught as the inverse operation of multiplication, they need not be introduced as a special topic that is difficult, requiring special treatment. A half will occur frequently and will represent the awareness that one rod is contained twice in another. When 8 is studied the fractions introduced will be , and , while the study of 6 will have ensured that and have also been met and used’. Fractions are therefore introduced at the very beginning of the Cuisenaire-Gattegno course because they demonstrate a particular relationship between two numbers, just as simply and clearly as in the case of addition, subtraction, multiplication or division of two numbers.

116

Chapter 6 Teaching The Basic Concepts Of A Fraction

The Study Of Fractions In this chapter, an understanding of certain concepts of a fraction will be developed. These ideas are important in themselves and because they give experience in the operation of partition division; in addition they serve as a basis for understanding other ideas, such as ratio and proportion. The concepts to be introduced are as follows: 1

fractions as relations between two numbers;

2 fractions as operators; 3 the equivalence of fractions.

Prerequisites For Work In Fractions Before fractions are introduced it is of course necessary for the pupil to understand 1

ordinal number—that is, to be able to count and put numbers in their correct sequence;

2 cardinal number—that is, to realize that a number may be considered as a whole or group; 3 equality—the relationship of equivalence, or equality, has of course already been understood by the pupils who have been handling the rods for a while.

117

The Cuisenaire Gattegno Method Of Teaching Mathematics

Fractions As Relations Between Two Numbers Fractions can be introduced from a discussion of such a pattern of rods as the one below.

FIG. 55. There are five white rods. If one is considered alone, the child sees that it is ‘one of the five’, which he can later call ‘one fifth’, written . If a new pattern is now studied, such as the one illustrated below,

FIG. 56. the child will still be able to say, even though the rods are larger, that any one of them is one fifth of the total length. By using other rods, the light green, the pink, and so on, he still sees as the relation of one length to another.

118

Chapter 6 Teaching The Basic Concepts Of A Fraction

Once the child has understood that one white rod is of a row of 5 white rods (Fig. 55), he will readily see that it is also of the rod representing that length, i.e. the yellow rod, or that the

relation of 1 to 5 is

. FIG. 57.

He should have no difficulty in adopting the pattern shown in the above illustration to represent the fraction . The fractions , , , , , , and introduced in the same way. It is important for the children to understand that each of these can be represented by a number of rod patterns; also the same wording should by used by the teacher each time: ‘Show me one third as the relation of the white rod to another length’; ‘Show me one third as the relation of the yellow rod to another length’;

119

The Cuisenaire Gattegno Method Of Teaching Mathematics

‘Show me one third as the relation of the orange rod to another length’.

Fractions With Numerators Greater Than One If, in the illustration in Fig. 55, two white rods are now considered, ‘two fifths’ can be introduced, written . The other fractions and can be taught in the same way and at the same time. Each fraction is illustrated with a variety of rods ; thus may be shown with yellow rods, black rods, blue rods and so on. The fractions , , , , . . . are all introduced in the same way ; presents no more difficulty than . The child moves easily from to to using white rods as follows:

FIG. 58a.

FIG. 58b.

FIG. 58c.

These fractions are studied using a wide variety of patterns and the child again sees fractions as relationships, not as fixed patterns.

120

Chapter 6 Teaching The Basic Concepts Of A Fraction

Other fractions such as , , and so on are similarly introduced, and the following type of questions can be asked: Show me

of the dark green rod;

Show me

of the brown rod.

The fact that these fractions are represented with two rods (e.g. a blue and a dark green for of 6) demonstrates that it is the relationship between two whole numbers which is being studied.

Gattegno’s Exercises In ‘Mathematics’ An expression such as × 5 = 1 leads to the concept of a fraction as an operator, operating on a quantity to produce another related to the first. Gattegno uses this concept extensively in his study of numbers to provide an understanding of partition division. It is introduced, in connection with the study of numbers up to five, at the same time as the other basic operations in Part IV, Book I (pp. 39-47) of Mathematics. The fractions are introduced in order and each in turn is dealt with in detail. Concentrating on one half, Gattegno gives a wide variety of exercises with half as an operator, including such examples as 3 – ( × 4) =, 2 + × 2 + × 4 = , and × 2 + × 2 = . He does the same with thirds before proceeding to fourths, and with fourths before proceeding to fifths. From the study of 6 onwards, he treats partition division as a

121

The Cuisenaire Gattegno Method Of Teaching Mathematics

mathematical concept to be studied along with the other basic operations. Each fraction is introduced through the medium of multiplication. From ‘two red rods equal one pink rod’ he obtains that a red rod is half a pink rod; from ‘two white rods equal one red rod’ it follows that one white rod is half a red rod; or, 2 × 2 = 4, thus 2 = × 4; and 2 × 1 = 2, hence 1 = × 2. Thirds, fourths and fifths are similarly obtained. (Mathematics, Part IV, Book I, exercises 19, 27, 30 and 35.)

Fractions As Operators 1

Using the white rod

Pupils learn that 1 = × 10, 1 = From the pattern below

× 7, 1 =

× 5, and so on.

FIG. 59. which reads 10 × 1 = 10, the inverse relationship 1 = derived.

122

× 10, is

Chapter 6 Teaching The Basic Concepts Of A Fraction

Pupils similarly learn that 3 = and so on.

× 10, 7 =

× 10, 9 =

× 10,

FIG. 60. Eventually, it would be an easy step to pass from 1 = × 5 and 5 = × 5 to 6 = × 5, 7 = × 5, 8 = × 5, and so on.

FIG. 61a 1 = 2

×5

FIG. 61b. 5 =

×5

FIG. 61c. 6 =

×5

Using rods other than the white

Pupils now learn that 5 = ×10 and 2 = fractions may be taken together.)

123

× 10. (These two

The Cuisenaire Gattegno Method Of Teaching Mathematics

One should always proceed from the pattern for multiplication, 2 × 5 = 10,

FIG. 62a. in order to arrive at 5 =

× 10.

FIG. 62b. The other product for 10, that is, 5 × 2 = 10 should be studied at the same time,

124

Chapter 6 Teaching The Basic Concepts Of A Fraction

FIG. 63a.

and its inverse relationship 2 =

× 10, arrived at.

FIG. 63b One can very quickly proceed to other rods; for example 4 = × 8, 2 = × 8, 3 = × 9, 2 = × 6 and so on, can all be studied in a very short time. By following the same procedure, it will then be a simple step to go from 2 = × 10 to 4 = × 10 and so

on. FIG. 64. 3

Reversing the procedure

125

The Cuisenaire Gattegno Method Of Teaching Mathematics

Up to this point pupils have been directed from multiplications to fractions. This procedure is now reversed, and questions of the type “What is of 5 equal to?’ are asked. This helps develop the concept of fractions as operators. From previous activities the pupil has some knowledge of how to proceed, but for each question the child should be left to work according to his own experience. For example, when asked to find × 7 = , he may have to place white rods alongside the black to reach this pattern:

FIG. 65. Or again, if he has to find × 10, and he does not realize that the red rod is one fifth of the orange rod, he will need to try various rods until he finds out that only if he chooses red rods will the requirement be met.

126

Chapter 6 Teaching The Basic Concepts Of A Fraction

FIG. 66. It is recommended that exercises be graded as set out in a and b above. The following questions can serve as a guide: 1

Using the white rod — 1

2

What is

× 10 equal to?

2 What is

× 10 equal to?

3 What is

× 5 equal to?

Using rods other than the white — 1

What is

× 10 equal to?

2 What is

× 10 equal to?

3 What is

× 8 equal to?

Because multiplications have already been widely practiced, it is at this stage that the close relation between multiplication and fractions should be stressed. The aim is to make the child read × 10 = 2 as naturally as he reads 5 × 2 = 10. His knowledge of the relations between the various operations will remain limited and he will not understand partition division in the form of × 10 = 2 unless he realizes that multiplication is closely related to fractions.

127

The Cuisenaire Gattegno Method Of Teaching Mathematics

To give the child mastery of the operators, he is made to use them situations. The following illustration situation from which he obtains 9 –

concept of fractions as in increasingly complex is an example of such a × 9 = 2 × 3 or 9 – × 9

= 3. FIG. 67. He can also be led, by various methods, to compare how fractions operate. For instance by means of this pattern

FIG. 68. he would be able to see that

×8=

128

×8=

× 8.

The Cuisenaire Gattegno Method Chapter 6 Of Teaching Mathematics Teaching The Basic Concepts Of A Fraction

Or he can be asked to discover which of a pair of expressions is the bigger (or the smaller), for instance: × 8 and × 10. When the child is capable of such activities it can safely be assumed that he knows how to use fractions in his problems.

Equivalence Of Fractions Gattegno in his books gives great importance to the concept of equivalence of fractions. Indeed, this concept forms the basis of his treatment of fractions with the Cuisenaire rods. The idea may be introduced to the class as follows: Show me

of the dark green rod (a white rod is produced).

Show me end).

of the dark green rod (two white rods end to

Can you replace these two rods with a single one equivalent to them? (this would be a red rod). If the red rod is now taken as the unit, can you show me dark green rod? (this would again be the red rod).

of the

From this point it is only a short step for the child to realize that is equivalent to , or equivalent to .

129 127

The Cuisenaire Gattegno Method Of Teaching Mathematics

After this game is repeated a few times, the child begins to acquire a stock of fractions which are equivalent to each other and to handle them with confidence. Another opportunity of giving the pupils experience with equivalent fractions occurs during the study of a particular number, say 12. The children could be asked questions such as: Can you show me a rod that is

of 12?

Is there a single rod which represents 12?

of 12?

of 12?

of

The children quickly accept the fact that of 12 and the concept of families of equivalent fractions is thus easily understood. At a later stage, other fractions can be added to the series, yielding finally =

=

=

=

=

=…

where the dots show that we could go on and on finding fractions which belong to this family. The importance of this approach is in the fact that it will make later work on irreducible fractions, operations on fractions,

130

Chapter 6 Teaching The Basic Concepts Of A Fraction

L.C.D. and other related topics much more readily accessible to the pupils. For instance, if the children are asked to find the answer to , they will know how to proceed.

+

Looking at the two relevant families of equivalent fractions (or at the corresponding rod patterns)

and

=

=

=

=

=

=...

...

=

they will easily be able to pick two fractions, one from each series, which possess the same denominator; in this case and The operation then becomes +

=

+

=

and the children carry it out with great confidence. It should by now be obvious to the reader that giving the pupils an early and varied experience with fractions and their families of equivalence will pay dividends during later work. It is strongly recommended that sufficient time be devoted to this aspect of the course.

Â

131

Chapter 7 The Study Of Numbers Up To 20

Some indication of the wide range of mathematical concepts which can be studied with the Cuisenaire rods has been given. Any teacher following the Cuisenaire-Gattegno course has the freedom to devise whatever situation he desires as an experience in addition, subtraction, multiplication or division. He is not restricted to any particular group of numbers for the demonstration of these concepts. Using the rods, he can just as easily demonstrate that seven fives equal thirty-five as he can that two fours equal eight. His aim is not the study of number facts but the use of number as a vehicle for the expression of these ideas which lead to the understanding of the four operations. Even when he limits his pupilâ€™s experiences to numbers from 1 to 10 he is not bound to use these numbers in any definite order. It will however be found most practicable to confine the pupilsâ€™ earliest experiences to small numbers and proceed gradually to

Â

133

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

larger ones. For this reason, in Chapter 5, discussion was limited to activities associated with rods no longer than the orange, that is, involving numbers up to ten. In that chapter, moreover, the main problem under consideration was the transfer of experience from names-names to number-names. The stage had not been reached of taking a particular number and considering ways of revealing to pupils the variety of relationships associated with it. This can now be done. With the study of numbers up to 20, one is not passing to a new stage of the course. The child should be led to see that the concepts he has formed and expressed previously apply equally to larger numbers. All the ideas needed for an understanding of arithmetic have already been introduced and the more thoroughly this has been done the less time will be needed now for the study of numbers 11 to 20.

Numbers May Be Studied In Any Order As before, there should be no attempt to organize or systematize work so that number facts are mastered. The aim is still the development of understanding; only incidentally is the knowledge of number facts acquired. Gattegno, in Book I of Mathematics, treats each number seriatim from 11 to 20 but also suggests elsewhere, that they may be introduced by either doubling or trebling from known numbers, e.g. 2, 4, 8, 16; 3, 6,

Â

134

Chapter 7 The Study Of Numbers Up To 20

12; 5, 10, 20; 7, 14; 9, 18; 5, 15. 11, 13, 17 and 19 would of course have to be introduced by some other method. Furthermore, it should be a simple matter for pupils to apply their knowledge of ordinal number and see that the length orange plus white can represent eleven, orange plus red, twelve, and so on. Place value need not be stressed as pupils can easily see that the numbers are broken down into one 10 and a number of units.

Study Of A Particular Number Exploration of the number By means of a mat for the particular number, for example 12, pupils read, write and discuss the wide variety of relationships contained in the various lines. With a different mat made by each pupil the teacher can gauge the depth of understanding achieved for each to the ideas presented. Any one line may be expressed in at least half a dozen different ways and there are usually additions, subtractions, multiplications and divisions in abundant variety. The reading and/or writing of these relationships is an important step towards developing the ability to play with figures, to think mathematically and, ultimately, to work mentally. In a given line of a mat for 12, the following are but a few of the expressions which can be given:

Â

135

The Cuisenaire Gattegno Method Of Teaching Mathematics

12 = 2 + 3 + 3 + 4

12 – (2 × 3) = 2 + 4

4 + 3 + 3 + 2 = 12

12 = (

× 12) + (

12 = (2 × 3) + 2 + 4

12 = (

× 12) + (2 × 3) + (

2 × 3 = 12 – (2 + 4)

6 = 12 – (

12 = (

× 12) + 2 + 4

× 12) + (

× 12) + (

× 12) × 12)

× 12)

× 12 = 12 – 2 – (2 × 3)

Although it is usual for pupils to supply most of the readings, the teacher can stimulate them by asking: ‘Who has the line ( × 12) + ( × 12) + ( × 12) + 1?’ or by saying: ‘Add to your mat the line 5 + ( × 12) + ( × 12) . Who can read (or write) this new line in a different way?’ It is to be realized that 12 has not been studied previously and that the fractions have not been met with in this context, but a large number of pupils will have developed sufficient understanding to give the more complicated readings, or writings. However, the main purpose at this stage is given in the original direction ‘make a mat for l2 and see how many ways of writing down each line you can discover’. Such a free exploration of the particular number serves as a general introduction and often gives a pointer to the teacher as to what aspects will need emphasis during later work. Addition and subtraction The pupils are asked to read and then write some of the lines they can see in their patterns. These could at first be written on the blackboard as the pupils read them out but erased before they begin their writing. For each line there are as many as

136

Chapter 7 The Study Of Numbers Up To 20

twelve different writings, all of which the children should be encouraged to find; for instance: 5 + 7 = 12 7 + 5 = 12 12 = 5 + 7 12 = 7 + 5

12 – 5 = 7 12 – 7 = 5 7 = 12 – 5 5 = 12 – 7

12 – 7 – 5 = 0 12 – 5 – 7 = 0 12 – (5 + 7) = 0 12 – (7 + 5) = 0

Each particular combination of 12 could be studied in this way. Addition with a gap Both additions and subtractions can be read or written by just looking at a pattern. This could be presented as follows: Form the length which represents 12 with your rods and put the black rod alongside. What must be added to 7 to make 12? or which rod makes up the difference between 12 and 7? Using the notation for addition, this could be written as 7 + = 12, 12 = 7+ . Translating written mathematical statements into rod patterns Practice in the reverse process of making rod patterns from written mathematical statements is an experience which should

137

The Cuisenaire Gattegno Method Of Teaching Mathematics

be given to the children. For example, pupils could be asked to make rod patterns for 9 + 3 = 12 12 – (8 + 4) = 0 etc.

12 – 4 = 8 12 – (6 + 5) = 1

8 + = 12

Completing given expressions in writing The final step in the study of the particular number would be the completion of expressions in writing, given with the instruction: ‘if you can’t do it, use your rods’. Pupils would be given such exercises as the following: 4+8= 12 – 3 = 4 + = 12 8 = 12 –

12 = 3 + + 5 12 – (3 + 4) = (2 + 3 + 7) – 10 = (5 + 7) – (3 + 9) =

Adding The Lengths Of A Handful Of Rods Another exercise related to our present topic and recommended by Gattegno deals with the various possible methods of adding a ‘handful of rods’ together. Although the scope of this exercise greatly increases when dealing with numbers larger than 20, it nevertheless helps at this stage towards developing the ability to

138

Chapter 7 The Study Of Numbers Up To 20

think mathematically and manipulate numbers for a special purpose. The activity, consists in finding the total length of a handful of rods in the easiest way, and should be treated as a game: 1

Take a handful of rods. Put them end to end and find their total length. (Generally, the addition would be done indiscriminately and unsystematically.) Jumble them together again. Can you do it now without actually putting them end to end? Try this several times with different handfuls.

2 Take a new handful and try to arrange the rods in groups whose lengths add up to ten, and add them in this way by groups of ten. Now take other handfuls of rods and add their lengths by groups of ten; see if you can do it by just looking at them. Is it easier than the way you did it before? 3 Now take a handful of rods and put rods of the same color into groups and add them in this way. Take other handfuls of rods and, just by looking at them, add their lengths together by first adding rods of the same color together. Is this the easiest way? The answer given to this last question is not important. The purpose of the exercise is to make the pupils realize that there are alternative ways of adding a series of numbers. Once

Â

139

The Cuisenaire Gattegno Method Of Teaching Mathematics

Â

introduced, the game can be repeated periodically and each child should use the alternative methods. He need not keep to the order given above, but could first try the method he thinks easiest and then check by using the other methods.

Using Staircases For Exercises In Counting The making of staircases with common differences was described in Chapter 5. It will be remembered that it was possible to make more than one staircase with a common difference of, say, three between each of the steps (3, 6, 9; 2, 5, 8; and 1, 4, 7, 10). At that stage we were concerned with ascribing values to the respective rods. These staircases can be used for more than simply relating values to rods, they can for instance be used to develop the ability to count in steps other than the unit. 1

Build a staircase with a common difference of four, starting with the white rod, until you pass twenty.

2 Look at your staircase; read the values going up (1, 5, 9, 13, 17, 21) read the values coming down (21, 17, 13, 9, 5, 1) 3 Shut your eyes; say the values going up your staircase;

Â

140

Chapter 7 The Study Of Numbers Up To 20

say the values going down; say the values going up and then down; 4 Make a number chart for four. 1 5 9 13 17 21

2 6 10 14 18 22

3 7 11 15 19 23

4 8 12 16 20 24

5 Count by four, starting from 9. Go backwards after you pass 20. Continue, in writing, the following series of numbers until you pass 20: 2, 6 . . .

Developing The Concept Of Common Differences Between Pairs Of Numbers This idea is of fundamental importance for the method Gattegno recommends for dealing with subtraction. Understanding it, he says, eliminates all the various mechanical difficulties associated with the process. This will be discussed in detail later, but for the moment it will be sufficient to consider the importance of understanding the terms ‘difference’ and ‘common difference’ during the early stages of the course.

141

The Cuisenaire Gattegno Method Of Teaching Mathematics

At this stage pupils should have no difficulty in equating questions of the type: ‘What must I add to the length of the pink rod to equal that of the blue?’ ‘Which rod makes up the difference between a pink and a blue?’ ‘What is B – p equal to?’ There are many ways of introducing the idea of common differences during the early study of numbers. Three such ways are indicated below. 1

Finding pairs of rods whose difference is common Show me a pair of rods that has a difference equivalent to three white ones. Can you find another pair which has the same difference? See how many different pairs of rods you can find with the common difference of three. Read out what you’ve found. Write down what you’ve found. Make rod patterns to show: 7 – 5 = 10 – 8;

15 – 10 = 13 – 8 . . .

Complete these expressions in writing:

142

Chapter 7 The Study Of Numbers Up To 20

8–3=6–

10 = 7 – 4

2 Reading or writing common differences in staircases Build a staircase with a common difference of a yellow rod between the steps. Start with the red rod. Who can read the first step of the staircase? (7 – 2 = 5). The last step? (e.g. 22 – 17 = 5). One of the other steps? (12 – 7 = 5). Who can read the whole staircase upwards? (5 = 7 – 2 = 12 – 7 = 17 – 12 = 22 – 17). Who can read the staircase both upwards and downwards? Try writing down what you see in each step. (7 – 2 = 5

12 – 7 = 5, etc.).

See exercises 3 to 5 in Part III, Book I of Mathematics. 3 Adding or subtracting the same value in each line to maintain the common difference 1

Place the dark green and pink rods side by side. Which rod makes up the difference between the two rods?

143

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 69. Place a white rod end to end with a dark green one and another white rod end to end with a pink. What does the rod pattern read now? (7 – 5 = 2).

FIG. 70. Add a yellow rod to each line. Does the difference remain the same? Can you read this new pattern? (12 – 10 = 2).

144

Chapter 7 The Study Of Numbers Up To 20

FIG 71. Does the difference remain the same whichever rod we add to both lines? Take away the white and the yellow rods and try any other rod. Try orange rods. Try two orange rods. 2 Can you tell me from memory the answer to 17 â€“ 9 = ? Set out a rod pattern for this.

FIG. 72. Add a white rod to each line of your pattern.

FIG. 73.

Â

145

The Cuisenaire Gattegno Method Of Teaching Mathematics

Is the difference still the same? Read the new pattern (18 – 10 = 8). Does this make 17 – 9 any easier as a subtraction? Change 15 – 9 = , 13 – 8 = , 15 – 7 = into easier equivalent subtractions.

Multiplication And Division (Including Fractions) 1

Make 12 using an orange and a red rod end to end. How many light green rods fit the length ? Do it with your rods and see if you are right.

FIG. 74. 4 × 3 = 12. How many different ways of reading this pattern can you find? Each child should be encouraged to see each of the following: 3 + 3 + 3 + 3 = 12 12 = 3 + 3 + 3 + 3

3 = ×12 × 12 = 3 146

Chapter 7 The Study Of Numbers Up To 20

4 × 3 = 12 12 = 4 × 3 How many 3’s in 12?

12 – (4 × 3) = 0 12 – 3 – 3 – 3 – 3 = 0

Some pupils may also see such relationships as 12 – 3 = 3× 3 12 – ( × 2) = 9

( × 2) + (2 × 3) = 12 6 = 12 – ( × 12)

Occasionally some readings could be written on the blackboard as they are read out but these should be erased before the pupils are asked to write down as many expressions as they can see in the pattern. 2 Make a train of four light green rods. How many pink rods will fit this train? Read what you can see in the pattern. Write it down. (4 × 3 = 3 × 4 or 3 × 4 = 4 × 3) What is the product of 4 and 3? of 3 and 4? What are the factors of 12? 3 Make a rectangle of four light green rods side by side. Now make a rectangle of three pink rods side by side.

147

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 75a.

FIG. 75b.

Place the pink rectangle on top of the light green rectangle. What does this show you? (4 × 3 = 3 × 4) 4 We can show this in another way by taking one rod in each rectangle to form a cross.

FIG. 76a. 4 × 3 = 12

FIG. 76b. 3 × 4 = 12.

Either of these crosses will remind you of the equal rectangles you formed above and also of what you can read from a train showing 3 × 4 = 4 × 3. Make the cross for 6 × 2 = 12 and write all that you can read from it. 6 × 2 = 12 12 = 6 × 2

2= 6=

148

× 12 × 12

Chapter 7 The Study Of Numbers Up To 20

2 × 6 = 12 12 = 2 × 6 How many 2’s in 12? How many 6’s in 12?

× 12 = 2 × 12 = 6 12 – (4 × 3) = 0

5 The final step is to find the color symbol for 12 on the wallchart and among the product cards; this will be described later. At the end of this stage pupils should have an understanding of eleven of the basic products up to 100, that is 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 and 20. Discovering odd and even numbers Which of the lengths from 1 to 20 (using the white rod as the unit) can be made with red rods only? Try the orange; the black; the orange and yellow end to end. Find out all the lengths which can be made up of red rods. Write them down. These are called even numbers. Those which cannot be made up of red rods are called odd numbers. Make a staircase showing the even numbers up to 20. Read them going up and then down.

149 147

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

Shut your eyes and read them. Make a staircase showing the odd numbers up to 21. (See exercises 37-39 in Part II, Book I of Mathematics). Discovering prime and composite numbers Let us look for rods which can be matched in length with trains of rods of one color onlyâ€”other than the white. Can it be done for the orange, the light green, the blue, the pink, the yellow rods?

FIG. 77. From this, pupils discover that some numbers can be decomposed into factors, while others cannot.

Â

150

Chapter 7 The Study Of Numbers Up To 20

Numbers that have factors are called composite numbers. Those that have no factors, other than 1 and themselves, are called prime numbers. Write down a few composite numbers that you know (or have found). Write down all the prime numbers up to 20. Here is how some composite numbers are formed: 6 = 2 × 3 and 3 × 2 4=2×2 12 = 4 × 3, 3 × 4, 2 × 6 and 6 × 2. Show how these composite numbers can be formed: 15 =

18 =

9=

Discovering remainders in short division Make a mat for 13, trying to use rods of one color only, and then complete the length with the appropriate rod. The mat can first be read as multiplications, for e× ample, 13 = 2 × 5 + 3. The pupils can then be asked: ‘How many 2’s in 13?’ (6, but one must be added, or, 6 with one left over). ‘How many 4’s in 13? 6’s in 13?’ and so on.

151

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 78. It is then pointed out that the rod ‘to be added’ or left over’ should be called a ‘remainder’. Thus in answer to the question, ‘How many 4’s in 13?’ either ‘3, remainder 1’, or ‘3, r. 1’ may be written. When it has been established that the mat can be read in two ways– ‘13 = 6 × 2 + 1’, and ‘How many 2’s in 13? 6, r. 1’, a new reading can be introduced: ‘13 divided by 2 equals 6 with remainder 1’. The notation for division 13 ÷ 2 = 6 r. 1 can now be used. After making, reading and writing mats for other numbers using both the notation for multiplication (15 = 2 × 7 + 1) and the notation for division (15 ÷ 7 = 2 r. 1), pupils should be given

152

Chapter 7 The Study Of Numbers Up To 20

experience in making patterns from written statements; for example ‘Make the pattern for 3 × 5 + 4 = 19; and for 15 ÷ 3 = 5’. Finally, expressions should be completed in writing, for example 3 × 5 + = 19

16 ÷ 3 =

153

17 ÷ 3 = + 2

Chapter 8 The Study Of Numbers Up To 100

A General Outline Of Activities Connected With Numbers Up To 100 The pupilsâ€™ grasp of ordinal numbers up to 100 should now be extended, either by using rods end to end as previously, or by using the system of crossed rods to represent multiples of numbers already learnt. There is usually little need to warn the child that he does not use a tan and a black rod end to end to represent 87. As many ideas as possible are presented using multiples often before numbers containing a units figure are introduced. After these have been studied, selected numbers are introduced by means of doubling or trebling from known numbers. These are

Â

155

The Cuisenaire Gattegno Method Of Teaching Mathematics

investigated as products of their factors—but not so intensively as the numbers up to 20. These numbers will also introduce further multiplication facts, since they are studied chiefly as products whose factors, once known, will provide for ease and confidence in multiplications and divisions. The pupils should be made to discover that the basic mathematical concepts, developed in connection with numbers up to 20, apply also to larger numbers. The child will now begin to use his mathematical knowledge to play with numbers, to manipulate them, perhaps just to see what happens but usually for some specific purpose. He should be encouraged to juggle with numbers and to use his mathematical knowledge to organize and control numbers freely. Most of this discussion will concentrate on the methods adopted for the basic operations with numbers up to 100. These will be seen to encourage the child to discard the rods and carry out the operations mentally, as soon as he feels ready for it. Some additional activities which develop the pupils’ ability to manipulate numbers will also be mentioned.

The Multiples Of Ten Up To 1oo As a first simple step, the pupil is asked to make trains using orange rods only. He knows that one orange rod is equivalent to

156

Chapter 8 The Study Of Numbers Up To 100

ten white rods and that 2 orange rods are 2 × 10 = 20 white ones, so there is little need to explain that 3 orange rods end to end are 3 × 10 = 30 white ones, and so on. He may soon discover that it is more convenient to place his rods side by side rather than end to end.

The Four Operations With Rods End To End (Or Side By Side) The introduction of these trains of rods is followed by a study of the four operations. The pattern outlined for studying a particular number (see Chapter 7) may be adapted to meet the following requirements: 1

the study should involve multiples of ten only, such multiplication and division facts as 30 = 3 × 10 and 3 = × 30, 10 = × 30 being examined (not, at this stage, 5 × 6 = 30 or 14 + 16 = 30);

2 the relationships between the numbers being examined should be emphasized. The following exercises indicate more clearly what is required: 10 + (4 × 10) =

× 60 =

(2 × 10) + (3 × 10) =

(

30 + 50 =

100 – 60 = 2 ×

100 – 30 =

70 — (

157

× 40) + = 50

× l00) =

The Cuisenaire Gattegno Method Of Teaching Mathematics

5 × 20 =

(3 × 30) – 60 =

80 ÷ 20 =

90 ÷ 40 =

Pupils are usually capable of manipulating their rods to complete such patterns as the above. They learn 1

to work from rod patterns to written symbols, answering the question, ‘What do the rods say?’;

2 to make a pattern with rods from the written symbols; for example, when asked to ‘show with the rods that 50 — 30 = 20’; 3 to complete an expression in writing (for example, 20 + 30 = 40 + ) by working from the written symbols to the rods to an answer and, when this is understood, from the written symbols direct to the answer; 4 to write sums from memory; for example, ‘make up a problem about the number 40’. It will be found that in most cases from now on a small amount of mental arithmetic is involved when the rods are used. For example, in the sum 30 + 50 the rods would be arranged in two groups, but when these are brought together there is no special rod giving the answer. The rods must be counted up to 80. Teachers should be aware of this slight change when the rods are used for the larger numbers. The shortage of rods, particularly orange rods, can be a handicap. With only ten orange rods to share between pupils, it is often not possible to work individually. Group work is always

158

Chapter 8 The Study Of Numbers Up To 100

possible, or additional rods can be acquired; supplies of separate rods are available from the Cuisenaire companies.

Carrying Out The Operations With Crossed Rods There is an alternative method of representing these larger numbers; they can be set out as products of their factors, in the manner described in Chapter 7. Thus, the light green rod placed across the orange rod would represent 30, the pink rod across the orange one 40, and so on.

FIG. 79.

The light greenâ€”orange cross representing 30.

This representation once understood, the same types of exercises as recommended above can be carried out with crossed rods. How each of the operations can be carried out by means of these crosses is described below.

Â

159

The Cuisenaire GattegnoChapter Method 8 Of Teaching Mathematics The Study Of Numbers Up To 100

1

Addition; for example 20 + 30 = 50. The orange-red and the orange-light green crosses are formed. If the orange rods are placed parallel to each other so that the two top rods are end to end with each other, the idea of addition can be conveyed. This technique, although very useful for a preliminary understanding of the operation, must not be taken too far, as the ultimate aim is to enable children to work it out mentally.

2 Multiplication; for example 3 × 20 = 60. Here again the orange-red cross is formed. The question is then asked: ‘Three such crosses would represent how many orange rods?’ The answer is 3 × 2 or 6. This can be shown with the rods by placing a light green rod on top of the red rod and replacing the light green-red cross by its equivalent: a dark green rod. The resulting dark green-orange cross represents the answer. 3 Subtraction; for example 80 — 50 = 30. By making the tan-orange cross to represent 80 and the yellow-orange cross 50, the difference between the tan and yellow rods is seen as a light green rod. The pattern is then read, ‘eight tens minus five tens equals three tens’, or 80 — 50 = 30. If necessary, the difference between the top rods can again be shown by placing the crosses close together. 4 Division. This operation should be confined at this stage simply to reading as divisions the various crosses

160 158

Chapter 8 The Study Of Numbers Up To 100

made with the orange rod. For example, by looking at the pink-orange cross and by removing one of the rods from the cross, pupils would understand that of 40 = 10 (by removing the pink rod), of 40 = 4 (by removing the orange rod), 40 ÷ 4 = 10, and 40 ÷ 10 = 4. Of course, pupils would have already become familiar with this from their experiences with the products of up to 20. Thus, by encouraging the pupils in each of the operations first to set out the crossed rods and then to discover the answer by inspection, progression towards finding the answers mentally without using the rods at all, is hastened.

Using The Rods To Represent Any Number Up To 100 To represent any number up to 100 (such as 47), pupils may choose one of two ways. 1

Rods may be placed end to end; thus four orange rods and a black rod end to end measures 47 white ones.

2 The pink-orange cross representing 40 can be made, with the black rod representing 7 placed nearby on the right. Once this technique is understood a broader scope for the four operations becomes available. Pupils should be left to discover

161

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

methods of manipulating the rods for themselves in order to find the answers.

Studying A Selected Series Of Numbers Activities with the selected series of numbers up to 100 provide an opportunity for furthering the pupils knowledge of multiplication and division. While they are being studied these activities should be supplemented with exercises in addition and subtraction. Special numbers are selected by a system of doubling or trebling from known numbers This system will enable pupils to develop their ability to manipulate numbers and will serve as an introduction to the remainder of the 37 basic products. Each number is therefore made interesting because of its factors rather than any of the other properties it may possess. The numbers selected may be treated in the following order: 20, 21, 22, 12, 25, 27,

Â

40, 80; 15, 42, 84; 63; 44, 88; 33, 24, 48, 96; 50, 100; 54; 63; 45, 162

30, 60; 66, 99; 55; 77; 36, 72; 90; 81;

Chapter 8 The Study Of Numbers Up To 100

14, 28, 6; 35, 70; 49; 16, 32, 64; multiples of 13, 15, 17 and 19.

A full treatment of how these numbers can be used is given in Gattegno’s Part I Book II of Mathematics.

Studying A Particular Number One way of introducing the factors of the particular number under study would be to adopt the following method, exemplified here for 40: 1

making patterns showing the factors: 40 = 4 × 10 = 10 × 4 = 5 × 8 = 8 × 5;

2 making trains showing that 5 × 8 = 8 × 5; 3 forming congruent rectangles showing that 5 × 8 = 8 × 5; 4 using the tan-yellow cross to represent 40; 5 showing the color symbol on the wallchart and product cards which represents 40; 6 expressing orally and in writing what can be seen in the tan-yellow cross, for instance that 5 × 8 = 40, 8 × 5 = 40, 40 ÷ 5 = 8, 40 ÷ 8 = 5, 5 = × 40, 8=

× 40;

163

The Cuisenaire Gattegno Method Of Teaching Mathematics

7 making crosses for 40 from oral or written direction; 8 completing in writing expressions concerning 40 and its relation to other numbers.

Discovering The Relationships Between The Numbers In A Particular Series This may be illustrated by using the particular series: 21, 42, 84; 63. After a detailed treatment of the basic number 21 as outlined above, the ideas discovered are associated as closely as possible first with 42, then with 84, and finally with 63. This can be done in the manner described below. 1

What is 21 doubled?

2 Make the light green—black cross. Double this product, forming a tower of three rods, red, light green and black. Can you see ‘twice times three sevens in this tower? 2 × (3 × 7) = 2 × 21 = 42 Can you also see ‘two threes times seven’? (2 × 3) × 7 = 6 × 7 = 42 Replace, in the tower, the rods representing ‘two threes’ with a dark green rod, thus making the cross for 42 (6 × 7).

164

Chapter 8 The Study Of Numbers Up To 100

Find the product 42 on the wallchart. 3 Other activities concerning 42 follow, such as making the trains 6 × 7 = 7 × 6; reading and writing sums from the cross; = or 42 ÷ 7 = , etc.;

of 42

reading and writing other sums from the tower, for example: 3 × (2 × 7) = 3 × 14 = 42; 42 ÷ 2 = 21 (by removing the red rod from the tower the answer, 3 × 7 is obtained); 42 ÷ 14 = 3 (by removing the red and the black rods the answer is obtained). 4 What is 42 doubled? Make the cross for 6 × 7 and use a red rod to show 42 doubled. What can you see in this tower? Pupils should easily see that 2 × (6 × 7) = 2 × 42 = 84 (2 × 6) × 7 = 12 × 7 = 84 They may have to rearrange their tower to see that (2 × 7) × 6 = 14 × 6 = 84

165

The Cuisenaire Gattegno Method Of Teaching Mathematics

5 An alternative method would be to reach 84 from 21 in the following way: Make the cross 3 × 7 = 21. Double 21, forming a tower. Double again by adding another red rod to make a tower of four rods. What can you see in this new tower? 2 × (2 × 3 × 7) = 2 × 42 = 84 (2 × 2) ×3 × 7) = 4 × 21 = 84 (2 × 2 × 3) × 7 = 12 × 7 = 84 Replace the two red rods in the tower with a pink rod. What can you see now? 4 × (3 × 7) = 4 × 21 = 84 (4 × 3) × 7= 12 × 7 = 84 General activities follow, with the tower, or towers, still in front of the pupils. Thus, with three separate towers for 84 on each child’s desk (84 = 2 × 6 × 7 = 2 × 2 × 3 × 7 = 4 × 3 × 7) pupils may be asked either to write as many expressions as they can concerning 84, or to carry out completion exercises such as the following: of 84 = 4 × 21 = of 84 +

of of 84 = 84 – 21 = of 84 = etc.

166

The Cuisenaire Gattegno Method Chapter 8 Of Teaching Mathematics The Study Of Numbers Up To 100

6 In a similar way 63 can be shown to be related to 21 but this time by means of trebling. What is 21 trebled? Find the answer to 21 + 21 + 21 = by using rods end to end. Make a cross to represent 3 × 7. Which rod would you use on top of this cross to make a tower equivalent to the treble of 21? What can you see in this tower? 3 × (3 × 7) = 3 × 21 = 63 (3 × 3) × 7 = 9 × 7 = 63 63 ÷ 7 = 3 × 3 = 9 63 ÷ (3 × 3) = 63 ÷ 9 = 7 etc. Make the cross for 9 × 7 or 7 × 9 and find its symbol on the wallchart. Complete these exercises in writing: of 63 + of 42 = 63 — 42 = of 4 × 21 — 3 × 21 =

of

Free Exploration Using Towers At the correct psychological moment, while interest in towers is at its peak, pupils should be asked to make other towers and to

165 167

The Cuisenaire Gattegno Method Of Teaching Mathematics

write down the sums they can see in them. Teachers will be surprised and delighted at the discovery of new products (and their factors) that pupils make from their crosses and towers.

More Systematic Study Of Numbers Up To 1oo The pupils have so far been studying the basic operations in relation to the multiples of ten up to 100 and multiplication and division in relation to the selected products on the wallchart. The present stage will serve to extend their experience of products and their factors. Some of the activities the pupils will meet with are not new. They have already been encountered in relation to numbers of up to 20. The present aim would be to show that the ideas then introduced apply equally to larger numbers. Dr. Gattegno gives some indication of these activities in exercises 27–40 of Part I, Book II of Mathematics. They include 1

making tables of products and factors;

2 finding the prime and composite numbers up to 100; 3 finding the multiples of numbers up to 100 including those of 13, 15, 17, and 19; 4 counting, both upwards and downwards;

168

Â

Chapter 8 The Study Of Numbers Up To 100

5 extending the study of odd and even numbers; 6 finding the numbers up to 100 that are squares; 7 finding common factors, prime factors, common prime factors, and highest common factors.

Teachers may, at first sight, be surprised at the inclusion of such items in an arithmetic course for second grade pupils but, with the ideas made accessible by means of the Cuisenaire rods and with the use of correct mathematical terms arising naturally from the very beginning, they provide an interesting contrast to the conventional but drab study of products and factors. The course is not restricted to drill in the basic multiplication facts to be used in fixed mechanical processes, rather the field remains open so that each number can be seen from every viewpointâ€”as odd or even, prime or composite, as a square, a cube, or some other multiple. The relationship between products can also be examined for common factors, common prime factors, or highest common factors. This detailed study of products up to 100 and their factors is, although revolutionary,much more beneficial than gaining the ability to multiply such a number as 279 by 4 by the use of some fixed and blind mechanical process. Here are some of the activities that may be carried out at this stage. 1

Â

Doubling and redoubling numbers; e.g. starting from 7 and going on doubling until 100 is passed;

169

The Cuisenaire Gattegno Method Of Teaching Mathematics

2 Halving numbers; e.g. starting from 96 and going on halving until a number smaller than 5 is reached; 3 Using doubling to multiply numbers by 4 ( = 2 × 2) and by 8 ( = 2 × 2 × 2); e.g. 4 × 16 = 2 × 2 × 16 = 2 × 32 = 64; 4 Using halving to multiply by 5 ( = 7 = of 10 × 7= of 70 = 35; 5 1

of 10); e.g. 5 ×

starting with 15 and finding an expression so as to obtain 100; e.g., 5 × (15 + 5) = 100; 100 = 15 × 2 + 70;

2 starting with 99 and finding an expression so as to obtain 12; e.g., 99 – 90 + 3 = 12; of 99 – 21 = 12; etc.; 3 writing sums from memory either freely about any number or concerning a given number (e.g. 72): 72 = 6 × 12 = 12 + 12 + 12 + 12 + 12 + 12 = 2 × 18 + 2 × 18 = 100 — 28 etc. 4 completing exercises in writing e.g. given 33 = 3 × 10 + 3 = 4 × 8 + 1, find: 33 ÷ 10 = 33 ÷ 4 = of (33 — 1) = of (33 — 3) = 5 solving such problems as

170

Chapter 8 The Study Of Numbers Up To 100

24 — ( × ) = 8 6 doing a variety of counting exercises; e.g. counting by 15, starting from 40, till 100 is passed; 7 adding the values of a handful of rods, either one by one, or in groups of equal numbers, or in groups which add up to 10; e.g.: 4 + 7 + 7 + 6 + 4 + 6 + 6 + 4 + 3 + 3 = 50 or (3 × 4) + (3 × 6)+(2 × 7) + (2 × 3) = 50 or (4 + 6) + (4 + 6) + (4 + 6) + (7 + 3) + (7 + 3) = 50

Manipulating The Rods For The Four Operations 1

Addition; for example, 23 + 35 + 19 = 1

Using rods end to end the following trains can be made: 0 + 0 + g read as 23 0 + 0 + 0 + y read as 35 0 + B read as 19

The almost automatic action is to collect all the orange rods first and put them together on one side, leaving the light green, yellow and blue rods. If these cannot be added mentally, they can be placed end to end and measured with orange rods. The new pattern will be

171

The Cuisenaire Gattegno Method Of Teaching Mathematics

six orange rods, plus B + y + g which is equivalent to o+b This can be written as 60 + 17. If necessary, the extra orange rod can be placed with the six others so that the answer can be seen as 77. This procedure can be set out either horizontally 23 + 35 + 19 = 60 + 17 or vertically

2 The numbers can also be represented in the following way: crosses, of which an orange rod is always part, for the ‘tens’ and single rods for the ‘units’. The pupils would thus obtain the following patterns: a red-orange cross and a light green rod a light green-orange cross and a yellow rod a white-orange cross and a blue rod

172

Chapter 8 The Study Of Numbers Up To 100

Once again the ‘tens’ are added first, either by inspection or by measuring, and are here found to equal 60. The three orange crosses can therefore be replaced by a single dark green-orange cross. As before, the units are then added mentally or measured, found to be 17, and the final answer (77) obtained from 60 + 17. It will be seen that this sequence of manipulations is an encouragement to work mentally. Pupils at this stage do not ‘carry’; they find the two totals and add them as whole numbers, thus changing a ‘hard’ addition into an ‘easy’ one. 2

Multiplication; for example, 4 × 24 =

Exactly the same procedure is followed for multiplication whether rods end to end or crosses are used. The pupils can see that by setting out four twenty-fours the operation expressed in writing would be 4 × 24 = (4 × 20) + (4 × 4) = 80 + 16 = 96 or, if set out vertically

or

173

3

The Cuisenaire Gattegno Method Of Teaching Mathematics

Subtraction; for example, 53 — 28 =

The methods employed for carrying out this operation are based on the concept of simplifying it into an equivalent difference. Thus 53—28 in one case might become 55 — 30 = 25 and in another 33 — 8 = 35 — 10 = 25 1 53 — 28 = 55 — 30 = 25 The trains which represent 53 and 28 are made up with the rods. Pupils know that if they add equal lengths to two trains they do not change the difference between their lengths. So, in this case, by adding a red rod to each train the lengths become 55 and 30. From these two lengths one can remove the length of three orange rods; what is left is the length 25. 2

53 — 28 =33 — 8 = 35 — 10 = 25

The rods are arranged as crosses and lengths to represent the two numbers. Looking at the crosses the children can see that difference between the yellow and the red rod is a light green rod, so both crosses (50 and 20) are removed and replaced by the single light green-orange cross (30); the pattern now reads 33—8. As before a red rod is added to each to make the pattern read 35—10, which can be easily solved.

174

The Cuisenaire Gattegno Method Chapter 8 Of Teaching Mathematics The Study Of Numbers Up To 100

The second method is the one to be encouraged for types which require no ‘borrowing’; for example, in the case of 76—43, by changing to the equivalent difference 36—3, the answer becomes obvious. However, no distinction is made between types. It will be realized that either method makes the subtraction easy enough to be done mentally. 4 Quotition division The shortage of rods provides occasional difficulty in carrying out division of numbers up to 100 if trains of rods are used to represent the numbers. This however should be possible if a full box is used, as the total length of rods of each color is one meter, or just under. Thus, short divisions like 99 ÷ 3 and 98 ÷ 2 would require the full complement of red and light green rods respectively. Pupils working in groups of four should be given at least one experience of these very long ‘short’ divisions. Teachers should also give some examples of ‘long’ division, having regard to the availability of orange rods. Such types as 50 ÷ 6 = and 49 ÷ 23 = can be managed with a full box. Such experiences have the effect of showing pupils that the operation simply means repeated subtraction and that ‘short’ and ‘long’ division are identical operations. On the other hand, pupils may discover a third method of representing numbers. To represent 47, for example, a child may use the dark green-black cross (42) with a yellow rod (5)

173 175

The Cuisenaire Gattegno Method Of Teaching Mathematics

nearby, or alternatively, a yellow-blue cross (45) with a red rod (2) nearby. With the rods arranged in this way, an important additional step towards understanding short division can be taken by the child. For instance, from the following expressions 6 × 7 + 5 = 47 or 5 × 9 + 2 = 47 he could be asked ‘How many 6’s, 7’s, 5’s or 9’s are there in 47?’ 47 ÷ 6 = 7 r. 5 47 ÷ 9 = 5 r. 2

47 ÷ 7 = 6 r. 5 47 ÷ 5 = 9 r. 2

In addition, he can now see clearly the fact that 47 — 5 = 6 × 7 = 7 × 6 and 47 — 2 = 5 × 9 = 9 × 5

176

Chapter 9 Gaining Automatic Response With Number Facts And Tables

Breadth of experience is a prerequisite for understanding the number system. Therefore, it is important to realize that the Cuisenaire material permits the child to gain this experience without the restrictions imposed when memorisation is required. Consequently, automatic response is not directly aimed at in this course. The development of understanding is considered to be of prime importance and hence, although much memorisation does in fact take place, it is viewed as a welcome by-product rather than the characteristic on which progress is based. However, many teachers feel that unless the child is receiving specific training in memorizing number facts no worthwhile learning is taking place. Where this view is held, it is generally found that the childâ€™s activities are restricted to some degree in

Â

177

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

order that the necessary drill may be given. As this is opposed to the basic theory of the Cuisenaire-Gattegno course regarding the development of understanding it is to be deplored. The answer to the problem lies in the realization that automatic response will come; that a certain stage is devoted to ensuring this; and that memorisation is not required until the child needs to use certain facts for more complex calculations. Too often the refusal to accept this results in the failure on the part of the teacher to distinguish between memorized facts and true knowledge. With the former, the child can only repeat what he has memorized (purely a conditioned response), while with the latter the child has a tool for thought or reasoning. Furthermore, if no particular emphasis is placed on working without the rods, the teacher has a valuable indication of the extent of the childâ€™s knowledge. For it has been found that when the child knows and understands something, he will, for that particular aspect, work without rods. Therefore, as the work progresses, the rods are discarded and a considerable amount of automatic response is incidentally obtained.

Automatic Response May Be Gained Concurrently With The Development Of Knowledge It is to be emphasized that the task of gaining automatic response with number facts and tables can proceed

Â

178

Chapter 9 Gaining Automatic Response With Number Facts And Tables

simultaneously with the task of developing the childâ€™s knowledge. Manipulative work with the rods over a considerable period teaches him that if efficiency and ease in working with numbers is to be attained, then automatic response is desirable. As the numbers grow larger and more complicated the child realizes this need more acutely. This occurs particularly during the study of numbers up to 100. With these thoughts in mind, then, what means could be adopted in order to achieve mastery of number facts and tables? For the sake of clarity, these discussions will be confined to the mastery of multiplication facts. Teachers can readily adapt the methods presented here to other topics. There is no doubt that the greater the variety of ways in which a given number fact can be presented to the child the greater the impression it makes on his mind and the more likely he is to recall it automatically when it is required. For this reason no one particular method will be stressed above all others. The Cuisenaire-Gattegno course does give a great variety of such approaches by presenting these facts, first through the medium of the rods, and secondly through the medium of the color symbols on the wallchart and product cards. Gattegno claims that these give sufficient experience to enable all the facts to be memorized incidentally. The discussions in this chapter will include:

Â

179

The Cuisenaire Gattegno Method Of Teaching Mathematics

1

Gaining automatic response by means of the Cuisenaire rods;

2 Gaining automatic response by means of the Cuisenaire wallchart and the product cards and counters; 3 Gaining automatic traditional methods.

response

by

means

of

Gaining Automatic Response With Tables By Means Of The Cuisenaire Rods There is no doubt that much automatic response is gained incidentally every time the rods are manipulated. The fact that reverse processes are continually being emphasized, that the child sees a given pattern as 4 × 3 = 12, 3 × 4 = 12, 4 × 3 = 3 × 4, 3 = × 12 and so on, helps him form durable mental associations for each fact. Again, from the green-yellow cross he has formed, the child can see that 5 × 3 = 15, 3 × 5 = 15 and to find 15 ÷ 3 = he simply removes the light green rod and looks at the remaining rod. Further, the teaching technique, recommended in an earlier chapter, of looking only at the space in a pattern and saying which rod would fit that space, or of hearing only the names of the rods in the pattern and giving the answer, should develop visual images which are major steps towards the memorisation of separate number facts.

180

Chapter 9 Gaining Automatic Response With Number Facts And Tables

Here are some additional exercises which present number facts from a variety of viewpoints. 1 Given two numbers, for example, 5 and 7, find their product by composition; that is, by placing the corresponding rods end to end to form trains and measuring them. Express the result in a cross to be read or written in as many ways as possible; for example, 5 × 7 = 35, 7 × 5 = 35; 35 ÷ 7 = 5; 35 ÷ 5 = 7; × 35 = 7; × 35 = 5. 2 Given the product and one factor, find the other factor; for example 42 and 6. This requires measuring 42 with the dark green rod, and then checking to see if six black rods fit. A different method of presenting this exercise would be to ask for × 42 = . 3 The teacher or a pupil calls a product and pupils make the appropriate cross. After various products have been called, pupils read them back and later write them down. 4 Forming decompositions of crosses; that is, changing crosses into towers: 1

For example, the pink-black cross representing 4 × 7, can be decomposed into a tower showing red across red across black, representing 2 × 2 × 7.

2 Such a tower can be rearranged in a variety of ways to show 2 × 2 × 7 = 2 × 7 × 2 = 7 × 2 × 2 = 4 × 7 = 14 × 2.

181

The Cuisenaire Gattegno Method Of Teaching Mathematics

3 From such a tower, pupils are asked to give the answers to divisions such as 28 ÷ 4, 28 ÷ 7, 28 ÷ 14, 28 ÷ 2. It will be realized that with a tower for a number like 60 a much longer series of divisions could be given: 60 = 2 × 2 × 3 × 5 = 4 × 3 × 5 = 12 × 5 = 4 × 15 = 10 × 6 = 2 × 30 Find:

60 ÷ 5 =

0÷6=

60 ÷ 15 =

60 ÷ 2 =

60 ÷ 12 =

60 ÷ 3 =

60 ÷ 4 =

60 ÷ 30 =

60 ÷ 10 =

5 Pupils can discover that 4 = 2 × 2, 8 = 2 × 2 × 2, 6 = 2 × 3, 9 = 3 × 3, 5 = × 10, 12 = 10 + 2 and 11 = 10 + 1, and that this would reduce the essential multiplication tables to those of, 2, 3, 7 and 10. This can be explained as follows, using the number 18. 1

Multiplying by 4 and 8 To multiply 4 × 18, 18 is doubled to 36 and redoubled to 72. To multiply 8 × 18, 18 is doubled to 36, redoubled to 72 and redoubled to 144.

2 Multiplying by 6 and 9 To multiply 6 × 18, 18 is trebled to 54 and the resulting number doubled to 108. Similarly for 9 × 18, 18 is trebled and trebled again. 3 Multiplying by 5

182

The Cuisenaire Gattegno Method Chapter 9 Of Teaching Mathematics Gaining Automatic Response With Number Facts And Tables

To multiply 5 × 18, 18 is multiplied by 10 to 180, and then this is halved to 90. 4 Multiplying by 11 or 12 To multiply 12 × 18, the fact that 12 = 10 + 2 is used. Thus, 12 × 18 = (10 × 18) + (2 × 18) = 180 + 36 = 216 Alternatively, using 12 = 3 × 2 × 2, 12 × 18 is 18 trebled to 54 which is doubled to 108, and redoubled to 216. To multiply 11 × 18, 18 is multiplied by 10 to 180 and then 18 is added, which makes 198. By playing with products and their factors in these ways, pupils develop the ability to manipulate the number system; they begin to think mathematically and become well acquainted with a large number of basic facts besides. It is however, desirable to ensure that all the pupils have acquired the mastery of multiplications at least up to the table of 12.

Gaining Automatic Response By Means Of The Wallchart And The Product Cards And Counters Gattegno claims that with this special equipment the basic multiplication facts, introduced by means of the rods, are finally clinched, and that in addition the need for drill methods is

183 181

The Cuisenaire Gattegno Method Of Teaching Mathematics

entirely eliminated. However, experience in the more normal methods of teaching suggests that this special equipment may not be absolutely essential—but the reader will be able to decide on this matter for himself. The Cuisenaire wallchart As previously explained, the wallchart contains the 37 basic products up to 100, that is, all the products up to 100 which have factors of 10 or less. These factors are represented by semi-abstract colored crescents, the colors of which correspond in each case with the colors of the crosses symbolizing the products.

FIG. 80a.

The symbol for 4

FIG. 80b.

184

The symbol for 8

Chapter 9 Gaining Automatic Response With Number Facts And Tables

FIG. 80C. The symbol for 16 The products are set out horizontally in a definite order, one being the double of the other (when read from left to right) or half of the other (when read from right to left). For example, the top line of products reads from left to right: 4

8

16

32

64

The method of depicting products like 16 with two pairs of factors (2 × 8 and 4 × 4) can be seen. The blank circle in the middle is for the counter on which the product is printed. This design provides the same stimulus to thought as do the crosses representing the products. Uses of the wallcharts

185

The Cuisenaire Gattegno Method Of Teaching Mathematics

1 The pupils first find the symbol which corresponds to each product as it is introduced. When found, it becomes closely associated with the cross representing the same product. The relation of each product to others on the same line of the chart can also be noted. In this respect, the wallchart is particularly valuable during the study of the selected series of numbers under 100. 2 In the early stages of the games with the product cards and counters, the teacher can point to the relevant product on the wallchart. Also in the early stages of the use of the chart, the blank circle may be lightly marked in pencil with the number the symbol corresponds to. 3 The reverse process to moving from trains of rods through congruent rectangles and crosses to the colored crescents might be attempted. If, for example, a particular color symbol on the wallchart is not known, the pupils could make the corresponding cross with the rods, form the rectangles and/or the trains, and on measuring the latter, name or write down the product. 4 With the wallchart in view, the teacher, or a pupil, can call out a product and the pupils name its factors. Alternatively, the teacher could call out a product and one factor, leaving the class to name the other factor. This particular game can also take the form of partition division. For example, find × 45, × 27, and so on.

186

The Cuisenaire Gattegno Method Chapter 9 Of Teaching Mathematics Gaining Automatic Response With Number Facts And Tables

5 Exercises dealing with the relationship between one product and another can also be tackled, for example: Find the double of 8, of 32. Find

× 64,

× l6,

of

of 32.

6 A more difficult task would be to discover the fractional relationship between products having a common factor. An example of this would be to relate 32 (pink-tan crescents), to 80 (orange-tan crescents); tan (8) being the common factor, 32 would therefore be read as of 80, and 80 and of 32. 7 Another purpose of the wallchart is to introduce pupils to the games with the product cards. The product cards The pack of product cards consist of 40 separate cards the size and texture of playing cards. On three of these, rules for four suitable games are printed, whilst each of the remaining 37 cards shows a symbol which corresponds to one on the wallchart. The material is flexible as it is not necessary to use the full pack for playing any game. Indeed, from the teachers’ point of view, this is a most attractive feature because the pack can be limited to include only those products which have already been introduced; these may then be progressively increased as new

187

The Cuisenaire Gattegno Method Of Teaching Mathematics

ones are studied. Playing the games using, for example, only the eleven basic products under 20 (4, 6, 8, 9, 10, 12, 14, 15, 16, 18 and 20) can be just as interesting as using the full pack. At times, too, it is possible to select a special group of products that are causing difficulty. 1 The main game is for two players, or two players with partners. The cards are shuffled and dealt and pupils hold their ‘hand’ so that they can see all the symbols on their cards. The cards are played in turn, one at a time, the object being to obtain tricks, consisting of both the cards played, and won by playing a product card whose number is bigger than the card played by one’s opponent. If for example, a card is played by one’s opponent which is larger than any card in one’s own hand, then the best procedure is to throw in one’s lowest product, keeping the larger products back to play against the opponent’s products. Scoring can be either by the number of tricks won or by taking the value of tricks won. In this game, it is better for only three cards to be dealt at a time to each of the two players. ‘Hands’ of three cards each are dealt until the whole pack is exhausted. Each of these could be called a ‘round’ and the scoring adapted for each ‘round’. Thus, if 18 cards form the pack, 6 rounds would make a complete game. 2 A second game is for a leader to select a card from the pack and show it to the group ; the child who recognizes the product and calls it out first, takes the card. This is repeated until the

188

The Cuisenaire Gattegno Method Chapter 9 Of Teaching Mathematics Gaining Automatic Response With Number Facts And Tables

pack is exhausted. The same alternative methods of scoring can be adopted. This is rather noisy in a classroom of 40 pupils but children like it. It is very satisfactory for odd groups in a corner of the classroom. 3 The teacher, or pupil, can call out the products one by one as revealed by successive cards in the pack, while pupils in the class or group make the corresponding crosses with their rods. This, of course, could be done with a selected series of products needing extra attention. Checking should be done by pupils reading back their products from the crosses in the order they were called from the pack. Variations on the games mentioned, and perhaps new ones, can easily be devised by teachers. The game of lotto, using the product cards and counters The pack of product cards allows for considerable flexibility in the game. The counters are simply small cardboard discs on which a number is printed. The counters fit neatly in the white circle between the colored crescents on the product cards. To play the game of lotto, one player who acts as caller of the counters, shuffles the product cards and deals them one at a time to each of three other players (or three groups of players) till the pack is exhausted, except, usually, for one card which remains hidden. Each player (or group) arranges his cards face

Â

189 187

The Cuisenaire Gattegno Method Of Teaching Mathematics

upwards and on his desk or on trays and the caller commences calling the counters one at a time. As each counter is claimed it is placed in the white circle of the relevant card. The winner is the player (or group) who is the first to have all his product cards ‘covered’ with the correct counters. The penalty for not claiming a counter when it is called, or for claiming the wrong counter, is to return one of the counters already won to the caller. The game is thus very similar to ‘Housie Housie’ or ‘Bingo’ except that the ‘numbers’ are handed to the player by the caller. It may be played either with ten players (three groups each of three players plus their caller); or with seven players (three groups of each two players plus a caller); or with 4 players (three individual players plus a caller). Again, it is not essential that the full pack of 37 cards be used thus; if the game were restricted to these 19 products: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 30, 40, 50, 60, 70, 80, 90 and 100, the caller would have the nineteen corresponding counters, each player (or group) would have six cards and one card would remain hidden. If the full pack were used, each player would have 12 cards and one would remain hidden. As the pack is shuffled after each game, players would receive endless variations in their ‘hands’ of 12 cards. In practice, of course, the pack usually consists of those products to which pupils have been progressively introduced.

190

Chapter 9 Gaining Automatic Response With Number Facts And Tables

Â Gattegno claims that pupils come to know all their products thoroughly by means of these multiplication games with the wallchart, the cards and the colored rods and there seems little doubt that the novel idea of associating color with value should be very effective in this regard. However, traditional methods of achieving the purpose should at least be considered.

Gaining Automatic Response By Means Of Traditional Methods Teachers who are following the Cuisenaire-Gattegno course but who would like to use conventional methods to gain automatic response with multiplication facts, should be aware of the change of circumstances under which their methods will be used. With conventional methods of teaching arithmetic, the pupil has been under constant pressure from the very beginning to learn his basic number facts. On the other hand with the Cuisenaire-Gattegno approach, the number facts have been met with and used many times with ease and confidence but compulsory memorisation has been completely ignored. The child has matured with his arithmetic and is gradually approaching full understanding of the basic mathematical concepts of pure number. When this stage is reached, usually just before the introduction of fixed procedures, the teacherâ€™s task is to help the child

Â

191

The Cuisenaire GattegnoChapter Method 9Of Teaching Mathematics Gaining Automatic Response With Number Facts And Tables

organize the number facts so that they will be conveniently classified ready for memorisation. In this respect the ‘number horizons method’ described (except for the name) by Cuisenaire in Numbers in Color is recommended. The number horizons method This method requires that a child, who is concentrating on automatic response, work simultaneously at three different levels, as follows: 1

In the upper horizon, where experimentation takes place and new number facts are investigated and explored;

2 In the middle horizon, where the child is gaining understanding of and familiarity with the number facts, concentrating on automatic response with them and memorizing them; 3 In the lower horizon, where the number facts have been thoroughly mastered and are being used with confidence and efficiency. In most classrooms, these three stages exist and proceed simultaneously at each level. At a given time in a certain child’s progress, he may be working in accordance with this chart:

190 192

Chapter 9 Gaining Automatic Response With Number Facts And Tables

FIG. 81. This particular chart indicates that the child is in the stage of automatic response with the numbers 1–20, of familiarization with the numbers between 20 and 30, and of classification of the numbers between 30 and 40. He may, by counting and limiting, have located numbers up to 100 and beyond. He occasionally ‘explores’ big numbers with his rods but, although his upper horizon is 100, he has built up his tables only up to 40. At the same time, he is using various techniques to gain mastery over the numbers between 20 and 30. This is the limit of his middle horizon. The numbers up to 20 have been studied and he understands them in every detail. Within the limits of this lower horizon his degree of mastery has led him to discard the rods and he uses these numbers with great confidence.

193

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

After a certain time, the horizons at each level have shifted so that the child now has mastery over numbers up to 30, is concentrating on numbers between 30 and 40, and is classifying numbers up to 50, although he may perhaps have commenced the exploration of numbers up to 1000.

FIG. 82 The horizons move in this way until all the number facts are covered. It is not necessary to always work in terms of groups of 10. Other convenient arrangements can be made. The organization of the multiplication chart shows the relation between the levels even while the tables are being built up. The

Â

194

Chapter 9 Gaining Automatic Response With Number Facts And Tables

child can be taught to read from the chart that 2 × 3 = 6, 3 × 2 = 6, 6 ÷ 2 = 3, × 6 = 3, × 6 = 2, and so on. In the building up of the tables, the teacher (or the child), can use either the form of the charts above, or the more usual table forms. Normal teaching methods to gain automatic response In ensuring automatic response all normal teaching methods may be used. There will be no need to elaborate on the following: 1

work with counting and number charts;

2 games with digit circles; 3 use of flash cards; 4 tables competitions.

195

Chapter 10 The Study Of Numbers Up To 1000—I

It would be useful at this point to mention briefly the main strands of this course, since somewhere during, or immediately after, the study of numbers up to 1000 these must be brought together in the culmination of the efforts made to develop in children an understanding of, and efficiency in, arithmetic.

Exploring The Number System During the study of the numbers up to 100, pupils were encouraged to use their mastery of these numbers in a series of games. They used the rods for such operations as 73 — 28 = and 5 × 17 = as a concrete introduction to such mental calculations such as

197

The Cuisenaire Gattegno Method Of Teaching Mathematics

73 — 28 = 75 — 30 = 45 5 × 7 = (5 × 10) + (5 × 7) = 50 + 35 = 85 They experimented with adding numbers, using handfuls of rods either singly or in groups. Later they added mentally by the method they discovered to be the easiest. For instance, they started with 100 and arrived at 15, at first by a simple operation such as 100 – 80 – 5 = 15, and later in a more roundabout way: 100 – (2 × 45) + (40 ÷ 8) = 15. While studying selected numbers, they were given experience with doubling and trebling to discover, among other things, that 42 is the double of 21 and that 3 × 7 is therefore half of 6 × 7. It will shortly be found that almost the entire course of study of numbers up to 1000 is designed to increase the pupils’ ability to manipulate the number system and to strengthen their capacity for mathematical thought.

Learning The Fixed Procedures All this has been done in preparation for the understanding of the fixed and systematic processes that will have to be employed for complex calculations. Gattegno, in Part I, of Book II of Mathematics, introduces these fixed processes immediately after the study of numbers up to 100, but he says elsewhere that he does so ‘merely as a concession to tradition’. This would seem to be the case because most of his exercises in Parts IV, V and VI

198

Chapter 10 The Study Of Numbers Up To 1000—I

of Book II of Mathematics aim at a detailed exploration of the numbers up to 1000 and not at the mastery of fixed processes. Certainly, to introduce the processes before the knowledge relating to the operations has been consolidated by further experience (such as can be gained with a study of the numbers up to 1000), could defeat the purpose of the method. It is for this reason that ‘the study of numbers up to 1000’ is discussed in this course before ‘discovering systematic procedures’. Nevertheless, at some time during the study of these numbers (preferably towards the end), the Cuisenaire-Gattegno course in arithmetic must be brought to a conclusion by the introduction of fixed procedures for addition, subtraction, multiplication and division. Very soon all the loose ends of this course must be brought together.

The Study Of Numbers Up To 1ooo However, the purpose of this chapter and the next is to concentrate on experiences with numbers up to 1000. Although the Cuisenaire rods can still be manipulated as the basis for these experiences, an even greater call is being made on the mathematical acumen of pupils. The technique used to develop the ability to work out, at first with the rods and then mentally, such types of problem as 17 — 9 = (numbers under 20) and 73 — 48 = (numbers under 100), is again followed for operations such as 710 — 289 = . Pupils

199

The Cuisenaire Gattegno Method Of Teaching Mathematics

gradually discover that what was previously done with smaller numbers can be repeated with the larger ones. The methods employed are still unsystematic, involving longer and more complicated manipulations, initially with the rods and later mentally, all this assisting towards increasing the child’s control of the number system.

A Plan For The Discussion Of Numbers Up To 1ooo Gattegno’s series of exercises in Parts IV, V and VI of Book II of Mathematics, deals almost exclusively with multiplication and division. The discussion here will involve an analysis of these exercises, but something must also be said of addition and subtraction. The following headings will therefore be used: in this chapter 1

Introducing the numbers up to 1000

2 Addition and subtraction of numbers up to 1000 and in chapter 11 3 Finding a store of useful products, involving a wide variety of experiences with multiplication and division.

200

Chapter 10 The Study Of Numbers Up To 1000—I

INTRODUCING THE NUMBERS UP TO 1000

The Notation For Numbers Up To 1ooo Which Are Multiples Of 1oo Forming ‘threes’ The device of forming ‘threes’, or towers of three crossed rods, serves as an effective introduction to the multiples of 100. Characteristically Gattegno uses this preliminary step as a means of providing additional experience in the manipulation of factors. In this way, pupils discover that 200 can be represented by a tower of two orange rods with a red rod on top; 300 is represented by two orange rods as the base with a light green rod on top, and so on. Manipulating ‘threes’ which represent known products under 100 The first step is to form a cross, for example a tan-orange cross, and to notice that if one factor is doubled the other must be halved in order to maintain the equivalence. Thus, if the orange rod is doubled (to 20 if the white rod is the unit), then the tan rod must be halved (to pink) for the cross to represent 4 × 20 = 80. On redoubling to four orange rods, the pink rod must be halved again in order to represent 2 × 40 = 80. Thus, with the three successive crosses, the following equivalent products have been illustrated: 8 × 10 = 4 × 20 = 2 × 40 = 80. The range could

201

The Cuisenaire Gattegno Method Of Teaching Mathematics

be extended one step further by halving the orange rod in the original cross and doubling the tan rod so that 5 × 16 could be included in the series. The second step in producing equivalent multiplications from crosses is to change them into a tower, at first using the red rod as one of the ‘three’, but later using any three factors. From known products, formerly shown by a cross, ‘threes’ are formed, thus: 24 = 6 × 4 = 6 × 2 × 2; 42 = 7 × 6 = 7 × 3 × 2; 90 = 9 × 10 = 9 × 5 ×2 The third step involves manipulation of these ‘threes’. Using, for e× ample, the two orange rods cross to represent 10 × 10, and changing this to 10 × 5 × 2, the child can see this to be equivalent to 10 × 2 × 5, 2 × 5 × 10, 5 × 2 × 10, 5 × 10 × 2 and 2 × 10 × 5. This, of course, should be repeated with other known products. Fourthly, the advantage of representing products as ‘threes’ for the purpose of division is experienced. By removing the top rod from a ‘three’, the product (or number) is being divided by the value of the rod removed. Thus, with 100 = 10 × 5 × 2, 100 ÷ 5 = 2 × 10 = 20; 100 ÷ 2 = 10 × 5 = 50; and 100 ÷ 10 = 2 × 5 = 10. Before leaving this section it must be mentioned that the formation of ‘threes’ in these preliminary exercises becomes the foundation for Gattegno’s method of making ‘hard’

202

Chapter 10 The Study Of Numbers Up To 1000—I

multiplications ‘easy’ by finding equivalent multiplications. This will be explained more fully later. Experience in the basic operations involving hundreds As previously with numbers up to 100, so here the study proceeds immediately to the basic operations with the new numbers introduced; no longer can rods be used end to end for this purpose—it becomes simply a matter of inspecting the top rods in the ‘threes’ concerned. For example: 1

300 + 500 = . The light green and yellow rods on top of the respective towers are seen to be equal in length to the tan rod, thereby suggesting the answer, 800;

2 500 — 300 = . The difference between the yellow and the light green rods is seen to be equal to the red rod, suggesting the answer, 200; 3 4 × 200 = . The four red rods are seen to be equal in length to the tan rod, providing the answer, 800; 4

× 900= . One third of the blue rod on top of the tower is seen to be equal in length to the light green rod, giving the answer, 300.

This is a simple but fundamental step, assuming greater importance when tens and units are added to the numbers concerned.

203

The Cuisenaire Gattegno Method Of Teaching Mathematics

Multiplying (And Dividing) By 1oo And By 1o Before numbers containing tens and units are introduced, opportunity is taken to demonstrate and teach multiplication and division by 100 and by 10. Using the three crossed rods for one of these numbers, pupils can see the answers. For example, the 7 × 10 × 10 ‘three’ can be easily read as 7 × 100 while 800 — 100 is obtained by removing the two orange rods from the tower 10 × 10 × 8 leaving the tan rod (8) as the answer. Indeed, by manipulating the ‘threes’ for this type of problem, the pupil soon finds he can discard the rods and find the answers mentally: 600 ÷ 60 = 80 × 10 =

500 ÷ 100 = 10 × 50 =

900 ÷ 90 = 6 × 10 × 10 =

The Notation For Numbers Which Are Multiples Of 10 With mastery over the easiest numbers up to 1000, pupils are introduced to the next group in two steps, first the multiples often up to 200 and then the remaining multiples up to 1000. 1

Studying the multiples of ten up to 200

204

Chapter 10 The Study Of Numbers Up To 1000—I

The first step is to include the orange rod in the representation of these new numbers as a ‘three’. The approach is made initially by finding the doubles of 60, 70, 80 and 90, so that the pupils concentrate for a while on the four numbers 120, 140, 160 and 180. If 180 is taken as an example, to the blue-orange cross a red rod might be added to form the ‘three’ 10 × 9 × 2. The equivalent multiplications are studied by arranging the rods in as many ways as possible, for example, 180 = 2 × 9 × 10 = 10 × 9 × 2 = 9 × 10 × 2 = 10 × 2 × 9 and so on, and a series of divisions can be given for either one or two of the rods to be removed from the ‘three’: 180 ÷ 9 =

180 ÷ 2 =

180 ÷ 90 =

Secondly, pupils discover that rods other than the orange can be used in a ‘three’ to represent these numbers; that is, 180 = 5 × 6 × 6 = 3 × 6 × 10 = 3 × 5 × 12, so that a new game may be played which involves seeing how many different ‘threes’ can be found to represent each of the four new numbers. In this way 160 is seen to equal not only 2 × 8 × 10 but also 4 × 5 × 8 and 4 × 4 × 10, and each of these expressions can be rearranged (4 × 5 × 8 = 8 × 5 × 4 and so on) by altering the position of the rods in the ‘three’. A still wider range for multiplications and divisions becomes possible: 35 × 4 = 5 × 7 × 4 = 10 × 7 × 2 (the last being a much easier multiplication). To find 140 ÷ 28 pupils may first represent 140 as 2 × 7 × 10, changing the ‘three’ to 4 × 7 × 5 and dividing by 4 × 7.

205

The Cuisenaire Gattegno Method Of Teaching Mathematics

The third step taken with the study of these four numbers (120, 140, 160, 180) is to find a second and entirely different method of representing them with the rods. 140, for example, can be represented by a cross of two orange rods, with a pink-orange cross nearby. The number is thus seen to be formed of 100 + 40 = 140 and the place value of the tens digit is demonstrated. This method of representing the numbers will be found suitable for addition and subtraction. Some exercises can be given at this point, keeping to the same numbers, and using the same method as was employed with numbers up to 100. Give through your rods the answers to 160 — 120 = or 40 + 80 = Then complete these expressions in writing: 120 — 80 =

80 + 60 =

and so on.

It is to be remembered that pupils should be encouraged to find their answers by inspection of the rods for addition, and by means of equivalent differences for subtraction. For instance 120 — 80 = 140 — 100 = 40 The remaining multiples of ten can now be introduced in a similar fashion by multiplying known numbers by 10 (for example, 10 × 19 = 190). At first pupils may experience difficulty in trying to express such numbers as ‘threes’; but they soon discover that 190 = 19 × 10, and that the ten can be formed as 5 × 2 providing them with 190 = 19 × 5 × 2. For 150, two ‘threes’

206

Chapter 10 The Study Of Numbers Up To 1000—I

can be found, 15 × 5 × 2 and 3× 5× 10, but for each of the others only one ‘three’ is possible. Once this difficulty has been overcome, exercises in the basic operations should follow. The aim, however, is still not to teach number facts but to develop understanding. The exercises should therefore not be of the kind: find

150 ÷ 5 = 5 × 38 =

but rather, after the pupils have given the rod patterns for the following 150 = 2 × 5 × 15 = 3 × 5 × 10 190 = 2 × 5 × 19 they should be asked find

of 150 = 190 ÷ 19 = 190 ÷ 5 = etc.

2

3 × 50 = of 190 = 5 × 38 =

Studying the multiples of ten up to 1000

The next step is to introduce the remaining multiples of ten. Pupils write down the products of 10 by all the numbers

207

The Cuisenaire Gattegno Method Of Teaching Mathematics

between 20 and 30 (210, 220, 230 . . . 290) 30 and 40, and so on, up to between 90 and 100. This comparatively large group of numbers gives abundant scope for investigation and exploration along the lines indicated above. By means of forming ‘threes’ with the orange rod and two other factors (540 = 6 × 9 × 10), by replacing the orange rod with the yellow and red rods (530 = 53 × 5 × 2), by using any three factors (560 = 14 × 8 × 5), or by forming ‘fours’ (560 = 8 × 7 × 5 × 2), a very wide range of multiplications and divisions becomes possible, for example: 540 ÷ 27 = 540 ÷ 180 = 540 ÷ 60 =

18 × 30 = 12 × 45 = and so on.

Teachers should realize that the formation of ‘threes’ for these numbers in many cases involves an understanding of the multiplications of numbers up to 100 which pupils are concurrently memorizing. The knowledge that 48 = 6 × 8 is used in the formation of the ‘three’ for 480 (6 × 8 × 10), and also in the ‘four’ for 960 (6 × 8 × 10 × 2), and can serve as a stimulus to their efforts to gain mastery of the 37 basic products. This stage seems to offer the opportunity for pupils to discover these new numbers for themselves. All that seems to be required is that pupils be invited to make ‘threes’ using the orange rod as a member of each ‘three’, and to write down as many sums as they can see in each. For example, from the dark green-blue-orange tower, they might obtain 6 × 90 = 540, ×

208

Chapter 10 The Study Of Numbers Up To 1000—I

540 = 60, and so on. Some, by changing their ‘three’ into a ‘four’ or an equivalent ‘three’ (540 = 6 × 9 × 10 = 2 × 3 × 9 × 10 = 18 × 3 × 10) might discover a wider variety of multiplications and divisions; for example, 20 × 27 = 540; 540 ÷ 18 = 30; and so on. It must be emphasized again that manipulation of numbers such as 540 are being used primarily to develop understanding of the operations. If a number fact is memorized, so much the better, but it is only of secondary importance at this stage. Completion exercises, therefore, are put in the following manner: Make the towers for 10 × 7 × 9 and 10 × 7 × 3 × 3. Complete these in writing: × 630 =

30 × 21 =

630 ÷ 70 =

630 ÷ 30 =

Representing these numbers as hundreds plus tens The alternative method of representing numbers which are multiples of ten, by showing the hundreds as a ‘three’ (with two orange rods as the base) and the tens as a cross (of which one of the rods is an orange rod), broadens the pupils’ experience both of the notation and of addition and subtraction. After a little practice with this representation (for example, 590 = (5 × 10 × 10) + (9 × 10) = 500 + 90) the usual routine may be followed:

209

The Cuisenaire Gattegno Method Of Teaching Mathematics

1

Making rod patterns from oral or written statements, e.g. 830 — 580 = 850 —600 = 250 or 830 — 580 = 330 — 80 = 350 — 100 = 250

2 Completing expressions in writing, using the rods as a prop, e.g. 830 — 580 = 3 Looking at an expression (e.g. 830 — 580 = ) and writing the answer; 4 Hearing the names of the numbers (830 — 580 = ) and writing the answer; that is doing the operation mentally.

Numbers Up To 1ooo Which Contain A Units Figure Introducing the units figure Up to this point pupils have been able to carry out the basic operations on numbers up to 100 and on numbers up to 1000 which were multiples of either 100 or 10. With the latter numbers, addition and subtraction have been carried out from rod patterns indicating the hundreds and tens, while multiplication and division have been experienced by manipulating the towers representing their factors. The method described below of introducing numbers up to 100 which contain a units figure also serves as an introduction to a

210

Chapter 10 The Study Of Numbers Up To 1000—I

new series of multiplication and division exercises, leading more directly than the exercises with ‘threes’ towards the formalisation of these operations. The pupil is told to start with 2 and to go on doubling until he reaches 1000, and is given help at critical points. He commences with 2, 4, 8, 16, 32, 64, and although he understands that the doubling of 64 would involve (2 × 60) + (2 × 4) = 120 + 8 = 128, he is told both how to read and write this number, being the first number over 100 with a units figure he has encountered. He is helped with his multiplication of 128 by 2 (2 × 128 = 2 × 120 + 2 × 8 = 240 + 16 = 256) and of 256 by 2. He finally arrives at 2 × 512 = 2 × 12 = 1000 = 24 = 1024 which is called ‘one thousand and twenty-four’. It is in this way that pupils gain experience in multiplying these comparatively large numbers by 2. After reaching 1024, they halve the numbers successively back to 2. Other series of doublings (and halvings) are attempted. ‘Start with 3 and go on doubling till you reach 1000. By starting with 7, with 9, with 11, and so on, skill at multiplying (and dividing) by 2 is developed. Incidentally, numbers such as 512 can be represented by a ‘three’ and practice with other multiplications and divisions is also obtained. For example, pupils should be made aware that 2, 4, 8, 16, 32, 64, 128 and 256 are all factors of 512, and thus realize that the numbers found by doubling can be the starting point for a score of useful products to be used in multiplications and divisions with relatively large numbers, in much the same

211

The Cuisenaire Gattegno Method Of Teaching Mathematics

way as the 37 basic products under 100 were useful with smaller numbers. By means of doubling, Gattegno introduces 1

the notation for the remaining even numbers up to 1000;

2 a series of exercises in multiplication and division designed to lead logically to an understanding of fixed procedures in these operations. This second aspect will be discussed in die next chapter. As regards the first, the notation for three figure numbers like 768 is analyzed, showing the place value of the digits. For example, 768 is shown with the rods to consist of 7 × 100, plus 6 × 10, plus 8. Thus, the rod pattern for this number would be: an orange-orange-black tower (700); an orange-dark green cross nearby on the right (60) and a little further to the right, a tan rod (8). In this way, the remaining numbers up to 1000 are introduced, and also through written exercises involving the place value of the respective digits (see exercises 15 and 16 in Part IV Book II of Mathematics). ADDITION AND SUBTRACTION WITH ANY NUMBER UP TO 1000 Pupils now have the means at their disposal for the completion of their study of the basic operations. This chapter will be

212

Chapter 10 The Study Of Numbers Up To 1000—I

concluded with a discussion of addition and subtraction of any number up to 1000. Here too, the study must remain informal if Gattegno’s theory is to be adhered to; this will be the last group of numbers to be studied before the fixed processes are introduced. As at every other stage of the course, this final step depends on the degree of mastery achieved by the child in the previous stages. 1

Addition

Here is a brief illustration of how the rods can be used to demonstrate addition and arrive at the corresponding written expressions. Example: 247 + 158 + 365 = 1

Make the appropriate rod patterns to represent the three numbers, keeping the towers (hundreds), crosses (tens) and the single rods in columns, thus: 2 × 10 × 10 + 4 × 10 + 7 1 × 10 × 10 + 5 × 10 + 8 3 × 10 × 10 + 6 × 10 + 5 As there is no fixed order for the addition, the path chosen to carry it out is unimportant. It may for instance be noticed that 240 (in 247) and 360 (in 365) total 600 and the rods can be rearranged to show this, thus helping with the remaining additions. The operation then becomes 600 + 100

213

The Cuisenaire Gattegno Method Of Teaching Mathematics

+ 50 + 8 + 7 + 5 = and the total can easily be obtained, 2 The three columns of rods can be inspected in turn and replaced by rods representing the respective totals. In the hundreds column, the light green, white (if used) and red rods are seen to equal the dark green rod; in the tens column, the rods on top of each cross are seen to total 15, while the rods in the singles column equal 20. The new pattern is (2 + 1 + 3) × 10 × 10

600

(4 + 5 + 6) × 10

150

7+8+5

20

In writing, this would be recorded as 247 + 158 + 365 = 600 + 150 + 20 = 770 or, if using the vertical notation

214

Chapter 10 The Study Of Numbers Up To 1000—I

A special exercise which should be attempted occasionally is to challenge the pupil to show as many different ways of doing an addition as he can and to decide for himself which is the easiest. For example, a child might provide the following expressions: 247 + 158 + 365 = (240 + 360 + 150) + (7 + 8 + 5) = 750 + 20 = 770 = (240 + 360) + 158 + 7 + 5 = 600 + 158 + 12 = 758 + 12 = 770 = (247 + 158) + 365 = 405 + 365 = 770 Gattegno argues that when exercises such as these are being included, the pupils attain quite a remarkable degree of mental agility for operating on numbers. 2

Subtraction

The same progression of steps as before is followed when dealing with subtractions. For this reason, there seems little need to discuss each of these steps in detail. The purpose here is to transform any given subtraction into an equivalent one which will make the operation easier to carry out. All that needs to be done is to show the pupils how to manipulate a wide variety of subtractions in this way. Here are some examples: 1

17 — 8 = 19 — 10 = 9

215

The Cuisenaire Gattegno Method Of Teaching Mathematics

2 78 — 53 = 28 — 3 = 25 3 73 — 28 = 75 — 30 = 45 4 101 — 97 = 104 — 100 = 4 5 860 — 370 = 890 — 400 = 490 or = 560 — 70 = 590 — 100 = 490 6 690 — 307 = 390 — 7 = 393 — 10 = 383 or = 693 — 310 = 393 — 10 = 383 7 687 — 254 = 487 — 54 = 437 — 4 = 433 8 587 — 294 = 387 — 94 = 383 — 90 = 393 — 100 = 293 or = 593 — 300 = 293 or = 583 — 290 = 593 — 300 = 293 9 806 — 378 = 506 — 78 = 528 — 100 = 428 or = 808 — 380 = 828 — 400 = 428 10 2004 — 789 = 2015 — 800 = 2215 — 1000 = 1215 or = 1304 — 89 = 1315 — 100 = 1215 If vertical notation was used, each transformation would be shown separately:

216

Chapter 10 The Study Of Numbers Up To 1000—I

Or

These various types of subtraction which normally require the adoption of at least three different techniques: borrowing, equi-addition and complementarity, have purposely been dealt with together. Under conventional methods, each type of subtraction would be taught separately. Gattegno argues that by developing an understanding of equivalent differences along the lines suggested, pupils not only have a single method which can be applied to any subtraction but they also have one which is so easy and efficient that they can accurately carry operations out mentally, and with numbers larger than 1000. In exercise 14 in Part VII of Book II of Mathematics, Gattegno writes: ‘Subtractions form families in that there are an infinite number of subtractions that show the same difference. Thus 5=5—0=6—1=7—2=8—3=... We can learn to work out any subtraction if we ask ourselves: “To which family does this pair of numbers belong;”. . .

217

The Cuisenaire Gattegno Method Of Teaching Mathematics

We must train ourselves to change a given difference into other equivalent ones, till we find one we can do at sight . . . Note that the only thing required is that each subtraction which you form be equivalent to the first one. You can make sure of this if you watch that you have either added or subtracted the same number from both the terms of your pair of numbers.’

Some Special Exercises In The Study Of Numbers Up To 1ooo Although it will be realized that much of what has been said above involves manipulation of the number system for some special purpose, Gattegno insists that children enjoy playing with arithmetic for its own sake and recommends they be given the opportunity to do so as often as possible. Here follows a list of exercises, many of which the pupil will have met during his study of numbers up to 1000: 1

Counting by 100’s to 1000 starting at 259; by 10’s, from 550 to 700; by 20’s starting at 190 to pass 300; by 50’s starting first at 10, then 20, then 30, then 40.

218

Chapter 10 The Study Of Numbers Up To 1000—I

2 Writing down all the multiples smaller than 1000 of 20, 30, 40, 50, 60, 70, 80, 90. 3 Starting at 900, finding different ways to get to 260. Starting at 170, finding different ways to get to 450. 4 Discovering a wide variety of facts about a particular number; for example: 360 = 6 × 60 = 30 × 12 = 2 × 180 = 4 × 90 = 180 + 180 = 100 + 260 = 400 — 40 = 1000 — 640 = × 720 5 Answering problems like 1000 — 290 = 220 + 720 — ( × ) = 120 6 Multiplying and dividing, for example: Given Find

468 = 3 × 6 × 13 × 2 18 × 26 = 6 × 78 = 13 × 36 = 468 ÷ 13 = of 468 =

7 Working with remainders in division: Given 475 = 3 × 12 × 13 + 7, find 475 — 7 = 36 ×

219

The Cuisenaire GattegnoChapter Method10 Of Teaching Mathematics The Study Of Numbers Up To 1000—I

8 Manipulating factors for easier multiplication; for example: 18 × 14 = 2 × 9 × 2 × 7 = 9 × 7 × 2 × 2 = 63 × 2 × 2 = 126 × 2 = 252 9 Doubling and trebling to find useful products for easier multiplication and division; for e× ample, we know that 896 = 8 × 112. Therefore 16 × 56 = 896; 896 ÷ 224 = 4; and 900 ÷ 224 = 4 r. 4. 10 Finding the squares of numbers and carrying out some interesting exercises with squares; for example, finding out that the sum of first four odd numbers=42. These exercises aim at developing the ability to think mathematically. Those involving the creation of mathematical statements about numbers, says Gattegno, are particularly valuable.

220 218

Chapter 11 The Study Of Numbers Up To 1000—Ii

In the previous chapter, the numbers up to 1000 were introduced and studied, thereby enlarging considerably the scope of activities. No practical difficulty associated with extending the range of additions and subtractions was encountered as the rods could be made to represent these numbers and in this way the concepts of addition and subtraction were put on even firmer ground. On the other hand, although it has been relatively simple to represent the multiples of 100 and of 10 as a ‘three’ containing orange rods, and hence to carry out certain fundamental multiplications and divisions, and although pupils should be now have mastered the 37 basic products, (a task, it will be remembered, that had been proceeding simultaneously with the activities outlined in Chapter 10), it is only to be expected that

221

The Cuisenaire Gattegno Method Of Teaching Mathematics

they would now experience a certain amount of difficulty in expressing at least some of the new numbers as a ‘three’ or a ‘four’. While 780 could easily be represented as 78 × 5 × 2, if it were not also recognized as 6 × 13 × 10, the addition of a units figure to make the number 783 would present problems which, for the present, the child would be unable to solve.

Exploring Numbers Of Three Figures Which Are Not Multiples Of 10 The way of overcoming this difficulty—typical of the Cuisenaire-Gattegno method—has already been met by the pupil during the earlier stages of his course. Right from the study of numbers of up to ten he was given complete freedom to explore the number system and to find his own way of expressing the relationship 7 — 4 = 3, for example, as of 14 — (20 — 16) = of 9. When he was introduced to the device of expressing a product and its factors by means of a pair of crossed rods he was invited to find for himself as many products as possible, using this technique. Similarly, during his search for new numbers, he used towers of three or more rods to represent their factors, which eventually brought him to the multiples of ten up to 1000. Now, at some opportune moment, he must be encouraged to discover for himself new numbers which can be represented by towers where

222

Chapter 11 The Study Of Numbers Up To 1000—Ii

any three or more rods are used. All that the teacher need say is, ‘Make a tower using any three rods and write down what you see in it’. Such freedom should bring him into direct contact with a host of new numbers, any one of which could be used to study multiplication and division. The fact that during his search the pupil will incidentally discover and remember the basic relationships concerning at least some of these new numbers, (for example, that 783 = 29 × 27) is of secondary importance at the moment, but will be most helpful at some later stage in his mathematical career when he happens to meet such a question as 790 ÷ 29 = . Then, says Gattegno, he will recall the number fact and will readily write that 790 ÷ 29 = 27, with 7 as the remainder. Thus, while the primary aim remains the development of mathematical concepts, particularly those relating to multiplication and division, mastery is, indirectly, being secured over at least some additional number facts, which Gattegno regards as most important because it enables pupils to carry out mentally some otherwise quite difficult operations.

Discovering Useful Products However, the pupil cannot be expected to explore the remaining numbers up to 1000 without some situations having been specially designed to stimulate his interest. Gattegno

223

The Cuisenaire Gattegno Method Of Teaching Mathematics

summarizes the pupil’s activities in this respect with the phrase ‘finding a score of useful products’, and encourages him by means of interesting experiences associated with the doubling and trebling of numbers, using tests of divisibility and building a table of squares up to 1000. These particular experiences are described in Part IV of Book II of Mathematics and will form the main topic for discussion in this chapter.

Doubling And Halving Numbers Up To 1ooo The pupil has already had some experience in doubling and halving numbers up to 100 and it has already been indicated* how he is assisted at particular hurdles, as when asked to ‘start with 2 and go on doubling till you reach 1000’. For example, it will be remembered that when he reached 256, he was shown the informal method of finding 2 × 256 = 2 × 250 + 2 × 6 = 500 + 12 = 512 Similarly, when asked to ‘go backwards till you reach the number you started with’, he was encouraged to treat each number in the series individually and to say how to find, for example, of 512: of 512 =

of 500 +

of 12 = 250 + 6 = 256

* See Chapter 10.

224

Chapter 11 The Study Of Numbers Up To 1000—Ii

Along with practice in doubling it was also shown that the pupil acquires the subsidiary experience of expressing any particular number in a series as a ‘three’ or a ‘four’. Thus, knowing that 512 = 2 × 256 = 4 × 128 = 8 × 64 = 16 × 32 he would understand such towers as 512 = 4 × 8 × 16 = 2 × 16 × 16 = 4 × 4 × 32 = 4 × 2 × 64 From these towers he could derive a wide variety of mathematical statements such as: of 512 = 4 × 32 = 128

512 ÷ 32 = 4 × 4 = 16

At this stage his experiences should be further extended to include remainders in division. It is important that he always be given the basic facts from which to make his deductions. At the moment he is simply developing an understanding of the operation, as a preparation for confident reckoning later. Thus, given that 520 = 16 × 32 + 8 = 8 × 64 + 8 = 4 × 128 + 8 he can easily find 520 ÷ l6 = 520 ÷ 64 =

520 ÷ 32 = 520 ÷ 128 = 225

The Cuisenaire Gattegno Method Of Teaching Mathematics

(520 – 8) ÷ 16 = 520 – (16 × 32) =

(520 — 8 ÷ 64 = 520 – (4 × 128) =

Multiplying And Dividing By 2 Teachers will find it essential to spend time developing the skill of doubling and halving until the pupil masters multiplication and division by 2 in the range of numbers up to 1000. It must be emphasized that he should not yet be shown any formal or fixed process for these operations. In exercise 14, Part IV of Book II of Mathematics, when reaching 96 in the series of doublings from 3, he is deliberately shown two informal methods: 2 × 96 = 2 × (90 + 6) = 2 × 90 + 2 × 6 = 180 + 12 = 192 = 2 × (100 – 4) = 2 × 100 – 2 × 4= 200 – 8 = 192 And when doubling 192 he is encouraged to say 2 × 192 = 2 × (190 + 2) = 2 × 190 + 2 × 2 = 380 + 4 = 384 Similarly, when halving 384, for example, he could possibly say that of 384 =

of 300 +

of 84 = 150 + 42 = 192

It is also to be noticed that Gattegno does not follow up these initial exercises in multiplying and dividing by 2 with a series of

226

Chapter 11 The Study Of Numbers Up To 1000—Ii

isolated numbers to be multiplied or divided (as occurs in our conventional teaching of arithmetic), but he consolidates the idea of repeated doublings, redoublings and halvings using new starting points. Exercises 17 and 18 include the following: Start with 5 and go on doubling. Start with 4, with 6, and go on doubling. Go backwards till you reach the number you started with. Start with 7, with 11, with 13, with 15. Go on doubling till you almost reach 1000.

Multiplying By 5 Pupils have already mastered multiplications by 10 and by 100 (when discovering the multiples of 100 and of 10), and, having now gained confidence with doublings and halvings, their skill is applied to other multiplications. They are reminded that the relation of 5 to 10 is , that 5 = of 10. They understand that the tan-yellow cross is equivalent to the pink-orange cross, that is, that 8 × 5 = 4 × 10. Therefore, says Gattegno in exercise 20, to multiply by 5, first multiply by

227

The Cuisenaire Gattegno Method Of Teaching Mathematics

10 and then halve the result. Pupils soon find that this can easily be done both with odd and even numbers: 5 × 18 = 5 × 17 = 5 × 33 =

of 10 × 18 = of 180 = 90 of 10 × 17 = of 170 = 85 of 10 × 33 = of 330 = 165

Owing to complications caused by remainders it is wise not to introduce division by 5 using this method. However, the alternative method of multiplying by 5 directly should be known: 5 × 33 = 5 × 30 + 5 × 3 = 150 + 15 = 165

Multiplying By 4, 8, 16, 32 And 64 The pupil’s skill in doubling may now be applied to isolated multiplications by 4 (2 × 2), 8 (2 × 2 × 2), 16 (24), 32 (25) and 64 (26). The pink rod in any tower can always be replaced by a red-red cross, thus: 4 × 73 = 73 × 2 × 2 = 146 × 2 = 292 Similarly, the tan rod may be replaced by a red ‘three’, allowing the answer to 8 × 67 to be found in the following way:

228

Chapter 11 The Study Of Numbers Up To 1000—Ii

8 × 67 = 67 × 2 × 2 × 2 = 134 × 2 × 2 = 268 × 2 = 536 This method can also be applied to multiplications by 16, 32 and 64. Again, in order not to restrict the pupil’s ability to a single technique, he should become acquainted with the alternative methods of multiplying by 4 and by 8: 4 × 73 = 4 × 70 + 4 × 3 = 280 + 12 = 292 8 × 67 = 8 × 60 + 8 × 7 = 480 + 56 = 536

Multiplying By 20, 40, 80 And By 50 Mastery having been obtained over doubling and over multiplications by 10, 2, 4, 8 and 5, it becomes a comparatively simple step to understand multiplications by 20, 40, 80 and by 50, making use of the corresponding ‘threes’ and ‘fours’. Thus: 17 × 20 = 17 × 10 × 2 = 170 × 2 = 340 = 17 × 2 × 10 = 34 × 10 = 340 17 × 40 = 17 × 10 × 2 × 2 = 170 × 2 × 2 = 340 × 2 = 680 = 17 × 2 × 2 × 10 = 34 × 2 × 10 = 68 × 10 = 680 17 × 80 = 17 × 10 × 2 × 2 × 2 = 170 × 2 × 2 × 2 = 340 × 2 × 2 = 680 × 2 = 1360 = 17 × 2 × 2 × 2 × 10 = 34 × 2 × 2 × 10 = 682 × 2 × 10 = 136 × 10 = 1360

229

The Cuisenaire Gattegno Method Of Teaching Mathematics

17 × 50 = 17 × 10 × 5 = 170 × 5 = 850

of 10 × 170 =

of 1700 =

Note that multiplication by 30, 60 and 90 is left until the ability to treble has been acquired.

Multiplication By 25 From earlier experiences pupils understand that 25 = of 100= of of 100. Their skill at halving can now be applied to multiplications of two-figure numbers by 25. Thus: 13 × 25 = = 325

or

of 13 × 100 =

of

of 1300 =

of 650

In exercise 25 of Part IV of Book II of Mathematics Gattegno suggests that this particular skill be applied to multiplications by 26, 27 and, perhaps by 23. Thus: 32 × 27 = 32 × (25 + 2) = 32 × 25 + 32 × 2 = + 64 = 800 + 64 = 864

of

of 3200

32 × 23 = 32 × (25 – 2) = 32 × 25 — 32 × 2 = 800 – 64 = 736

230

Chapter 11 The Study Of Numbers Up To 1000—Ii

Making Use Of The Ability To Double It will be noted that by first developing the ability to double and halve, pupils acquire a useful store of products, meeting many ‘threes’ and ‘fours’ from which they see a wide variety of multiplications and divisions (including divisions with remainders); they also apply their skill to isolated multiplications by 2, 4, 8,16, 32, 64, by 5 and 25. At the same time as carrying out these activities, pupils gain experience in alternative methods of multiplying, which, though informal, lead towards a fuller understanding of the operation. The next step is to develop skill in trebling, adding further to the store of useful products and, incidentally, extending the pupils’ experiences with multiplications and divisions in general and with multiplication by 3, 6, and 9, and by 30, 60 and 90 in particular.

Trebling Up To 1ooo And Dividing Numbers By 3 Teachers will realize the difficulties associated with trebling and dividing by 3; until the Cuisenaire-Gattegno course is understood, teachers may have to decide whether to develop the skill or simply to indicate its possibilities to the more advanced pupils. Where pupils have mastered all previous steps of the course, these activities should cause no undue anxiety.

231

The Cuisenaire Gattegno Method Of Teaching Mathematics

Exactly the same procedure as for doubling and halving should be followed and practiced extensively before proceeding with the applications. This is illustrated by the examples that follow: Start with 2 and go on trebling till you almost reach 500. Start with 13 and go on trebling till you pass 1000. Go backwards till you reach the number you started with. As before, pupils are assisted over the initial hurdles: 3 × 117 = 3 × (110 + 7) = 3 × 110 + 3 × 7 = 330 + 21 = 351

Using Numbers In Any Series Of Trebles For The Study Of Multiplication And Division As an illustration, the following example is used: 972 is the last number under 1000 in the series trebling from 4: 4, 12, 36, 108, 324, 972. Find some ‘threes’ concerning 972 and write the sums you can see: 972 = 3 × 3 × 108 = 3 × 9 × 36 = 6 × 6 × 27 = 9 × 9 × 12 = . . . of 972 = 3 × 108 = 324 972 ÷ 36 = 3 × 9 = 27 12 × 81 = 972

232

Chapter 11 The Study Of Numbers Up To 1000—Ii

etc.

Studying Remainders In Division Given that 980 = 27 × 36 + 8 Find

980 ÷ 27 = (980 – 8) ÷ 27 =

980 ÷ 36 = 980 – ( × ) = 8

Applying The Ability To Treble To Multiplications By 6 And 9 As the dark green rod in a cross can be replaced by a red-light green cross (6 = 2 × 3), and the blue rod by a light green-light green cross (9 = 3 × 3), multiplications by 6 and 9 can be carried out as follows: 43 × 6 = 43 × 3 × 2 = 129 × 2 = 258 =43 × 2 × 3 = 86 × 3 = 258 47 × 9 = 47 × 3 × 3 = 141 × 3 = 423 Alternative methods of multiplying must also be studied and compared; pupils should decide for themselves which is the easiest method: 58 × 6 = 58 × 3 × 2 = 174 × 2 = 348

233

The Cuisenaire Gattegno Method Of Teaching Mathematics

= 58 × 2 × 3= 116 × 3 = 348 =6 × (50 + 8) = 6 × 50 + 6 × 8 = 300 + 48 = 348 =6 × (60 – 2) = 6 × 60 – 6 × 2 = 360 – 12 = 348

Multiplying By 30, 60 And 90 As in doubling, pupils express the multiplication as a ‘three’ or a ‘four’ always using the orange rod as the base: 30 × 37 = 27 × 3 × 10 = 81 × 10 = 810 60 × 27 = 27 × 6 × 10 = 27 × 3 × 2 × 10 = 81 × 2× 10 = 162 × 10 = 1620 90 × 27 = 27 × 9 × 10 = 27 × 3 × 3 × 10 = 81 × 3 × 10 = 243 × 10 = 2430

Using Doubling And Trebling For Rapid Multiplications It will be noted that the skills developed above concerning doubling and trebling are designed to allow for greater facility in mental calculations. In the Cuisenaire-Gattegno course no distinction at all is made between ‘short’ and ‘long’ multiplication, or between ‘short’ and ‘long’ division. The operations are known simply as multiplication and division.

234

Chapter 11 The Study Of Numbers Up To 1000—Ii

Incidentally, mastery of doubling and trebling techniques reduces the essential multiplication tables to those of two, three, seven and ten, those of four, five, six, eight and nine being obtainable from them. However, as has been indicated above, it is not recommended that knowledge of all the tables be neglected. They are all included among the 37 basic products and provide the pupil with essential alternatives to doubling and trebling. Thus, although answers have been restricted for the most part to numbers up to 1000, pupils are now equipped to multiply by 2, 3, 4, 5, 6, 8, 9,10, by 25, 26, 24, 27,23, by 20,30,40, 50, 60, 70, 80, 90 and 100. It remains now to establish methods of multiplying by 7, by 11, and by 12 and then to develop an understanding of multiplying by any two-figure number.

Multiplying By 7 For multiplication by 7, it is necessary to know the seven times table; the alternative informal methods are used: 7 × 58 = 7 × (50 + 8) = 7 × 50 + 7 × 8 = 350 + 56 = 406 = 7 × (60 – 2) = 7 × 60 – 7 × 2 = 420 – 14 = 406

235

The Cuisenaire Gattegno Method Of Teaching Mathematics

Multiplication By II For multiplication by 11, pupils are encouraged to use 11 as 10 + 1 and say, ‘ten times the number, plus the number’: 11 × 43 = 10 × 43 + 43 = 430 + 43 = 473

Multiplication By 12 Pupils use 12 = 10 + 2 and say ‘ten times the number, plus double the number’: 12 × 43 = 10 × 43 + 2 × 43 = 430 + 86 = 516

Multiplying Two-Figure Numbers Together This is again a matter of using and extending skills already developed. First, for numbers easily factorized, (for example, 43 × 14 = ; or 18 × 26 = ) the operations should be set out as a ‘three’ or a ‘four’ and the factors (or tower) manipulated in the best possible way, thus making the multiplication easy to do mentally: 43 × 14 = 43 × 7 × 2 = 301 × 2 = 602

236

Chapter 11 The Study Of Numbers Up To 1000—Ii

18 × 26 = 2 × 9 × 2 × 13 = 13 × 9 × 2 × 2 = 117 × 2 × 2 = 234 × 2 = 468 Second, for numbers not easily factorized, or for prime numbers, there are several ways of transforming the operation, which the pupil will have already met and understood. For instance: 29 × 34 = 34 × (30 – 1) = 34 × 30 – 34 = 1020 – 34 = 986 29 × 34 = 34 × (20 + 9) = 34 × 20 + 34 × 9 = 680 + 306 = 986 29 × 34 = 29 × (30 + 4) = 29 × 30 + 29 × 4 = 870 + 116 = 986 The vertical notation may be preferred, in which case the above calculations would appear as

or

or

237

The Cuisenaire Gattegno Method Of Teaching Mathematics

The problem here for the pupil is to understand the distributive property of multiplication with respect to addition; this can easily be demonstrated with the Cuisenaire rods. In exercise 30 in Part IV of Book II of Mathematics, Gattegno uses the blue-black cross (representing the product 9 × 7) to illustrate this property. The rectangle made of seven blue rods can be decomposed into two rectangles of seven dark green and seven light green rods respectively, since we know that the blue rod is equivalent in length to the dark green rod + the light green rod. The rectangle made of nine black rods can similarly be decomposed to one rectangle of nine red rods and one of nine yellow rods. Pupils see from the final rectangles that: 9 × 7 = (6 + 3) × (5 + 2) = (6 × 5) + (6 × 2) + (3 × 5) + (3 × 2) = 30 + 12 + 15 + 6 = 63

Other Means Of Finding Useful Products At this stage, pupils have been engaged in exploring the number system up to 1000. They have been helped to discover those numbers that are related as doubles or trebles and, in so doing, have developed and extended their understanding of multiplication and division.

238

Chapter 11 The Study Of Numbers Up To 1000—Ii

But they are still not ready for the fixed processes. There are further sources of supply for useful products. These include 1

some special products easy to remember;

2 knowing and using tests of divisibility for various numbers; 3 knowing and using squares. 1

Some special products

If it is known that 3 × 37 = 111, it will be easy to remember that 6 × 37 = 222; 9 × 37 = 333; 24 × 37 = 888, and so on. 2 Knowing and using the tests of divisibility for various numbers Pupils already know, and have frequently used, the tests for divisibility by 2, 5 and 10. They should now be encouraged to discover for themselves by writing down and observing multiples of 3 (and the related numbers 6 and 9) that a number is divisible by 3 if the sum of its digits is divisible by 3. The tests for 4, 8 and 25 are also useful to have in reserve, and they could discover the test for divisibility by eleven by an examination of its multiples. In this way they are able to recognize a three-figure multiple of eleven by the fact that the middle figure is always the sum of the other two (attention being given to cases where carry over is involved). For example: ‘are these numbers multiples of 11: 913, 462, 374, 792, 638, 627, 363? If they are, give their factors’.

239

The Cuisenaire Gattegno Method Of Teaching Mathematics

Pupils are next led to recognize the multiples of 22, 44, 88; of 55 and of 33, 66, and 99. Finally, pupils are asked to multiply mentally such products as

or

32 × 33 = 32 × 3 × 11 = 96 × 11 = 1056 16 × 66 = 16 × 2 × 3 × 11 = 48 × 2 × 11 = 96 × 11 = 1056

It is admitted that in our normal curriculum for mathematics the work which has just been described is usually left for pupils in secondary grades. Teachers have to find out for themselves, by following the sequence of exercises described, whether pupils in lower primary grades can master such ideas. If the reader is already startled by what is normally regarded as difficult arithmetic for third and fourth grade pupils, he will be further astonished at the third source of useful products. 3

Knowing and using squares

The notion of squares has already been introduced in the study of numbers up to 20 and up to 100. Pupils already know, for example, that a square made of four pink rods is called ‘four squared’ and written 42. They have also measured the four pink rods end to end and discovered that 42 = 4 × 4 = 16. Through the discovery of squares up to 100, the idea has been suitably developed for discovering squares up to 1000. These would be squares extending from 12 to 312.

240

Chapter 11 The Study Of Numbers Up To 1000—Ii

With the object of carrying out the operation mentally, Gattegno analyses squares into 1 Squares which can easily be found using previous knowledge These give no difficulty, as pupils already know the squares of numbers up to 10 and can easily find 202 and 302. 2

Squares easy to work out from previous knowledge 1

Even numbers squared are multiples of 4; e.g.: 182 = 22 × 92 = 81 doubled and redoubled to 324.

2 Multiples of 5, when squared, are multiples of 25; e.g.: 152 = 52 × 32 = 25 × 9 = × 900 = × 450 = 225 3 Multiples of 4, when squared, are multiples of 16; e.g.: 242 = 42 × 62 = 36 doubled and then this answer redoubled three times to 576. 4 Multiples of 7, 8, 9 and 11 can easily be decomposed into their factors; (e.g. 222 = 112 × 22 = 121 × 4 = 484). Thus, the majority of squares less than 1000 are easy to work out using their factors and the squares of their factors. In the ‘table of squares’ that pupils are building they would now have all the squares of numbers up to 10 plus 122,142,152,162,182, 202, 212, 222, 242, 252, 272, 282, and 302. All of these can be worked out mentally.

241

The Cuisenaire Gattegno Method Of Teaching Mathematics

3

The remaining squares

How can the squares 132, 172, 192, 232, 262, 292 and 312 be found mentally when required? One method, of course, would be to multiply the number by itself but Gattegno proceeds to show our lower primary pupils a ‘special and simpler way’, one which involves the presentation and use of ideas normally left for secondary schooling. These may readily be recognized in their algebraic form as 1

(a + b)2 = a2 + b2 + 2 ab

2 (a – b)2 = a2 + b2 – 2 ab Gattegno, however, applies them to the arithmetic involved in such squares as 1

312 = (30 + 1)2 = 302 + 12 + 2 × 30 × 1

2 292 = (30 — 1)2 = 302 + 12 — 2 × 30 × 1 3 232 = (20 + 3)2 = 202 + 32 + 2 × 20 × 3 How can the Cuisenaire rods make ‘simple’ what has proved to be too difficult for primary grade pupils in the past? It can be first demonstrated with the rods that 92, which is equal to (4 + 5)2, is also equal to 42 + 52 + 2 × (4 × 5). After repeating this exercise with other smaller squares, pupils should be able to apply it to squares such as 312. First 92 is made using nine blue rods placed side by side. On top of this square and in one corner 42 is placed, made with four

242

Chapter 11 The Study Of Numbers Up To 1000—Ii

pink rods, and in the opposite corner 52, made with five yellow rods side by side. In the remaining spaces, which are two equal rectangles, either four yellow rods or five pink rods are placed.

FIG. 83. Once the equal rectangles have been measured and found to be either 4 × 5 or 5 × 4, the rods in the rectangles may be removed to allow the blue base of 92 underneath to set the rectangles into sharper relief. An attempt should then be made to read the pattern. The blue square underneath, (9)2, is seen to be equal to (4 + 5)2. The two equal rectangles are each 4 × 5, or its equivalent 5 × 4. Bearing these two facts in mind, the patterns can be read:

243

The Cuisenaire Gattegno Method Of Teaching Mathematics

92 = (4 × 5)2 = 42 + 52 + 2 × (4 × 5) Pupils should now make the patterns for other squares; for example, 52 = (2 + 3)2 = 22 + 32 + 2 × (2 × 3), or any other square of side less than 10 for which single rods may be used. When properly understood, the idea can be applied to find any one of the squares with which we have been dealing; for example: 132 = (10 + 3) 2 = 102 + 32 + 2 × (10 × 3) = 100 + 9 + (2 × 30) = 109 + 60 = 169 or 232 = (20 + 3)2 = 202 + 32 + 2 × (20 × 3) = 400 + 9 + (2 × 60) = 409 + 120 = 529 In a similar way, the relationship involved in (a — b 2 = a2 + b2 — 2 ab can be discovered, for example using 42 and representing it as the square of a difference, say (9 – 5)2.

244

Chapter 11 The Study Of Numbers Up To 1000â€”Ii

If a pink square is made and on two adjacent sides two rectangles are placed, each formed of five blue rods side by side, the total area produced can readily be seen to equal 92 + 52: by covering the protruding part of the figure with five yellow rods, an area equal to a blue square is produced.

FIG. 84.

Â

245

The Cuisenaire Gattegno Method Of Teaching Mathematics

If the two blue rectangles (2 × 5 × 9) are subtracted from this total area, the pink square (42) remains. In writing this would be 92 + 52 – 2 × (5 × 9) = 42 When (9 – 5)2 is written instead of its equivalent 42, the equation then becomes (9 – 5)2 = 92 + 52 – 2 × (5 × 9) Once demonstrated, the exercise must be repeated using other rods; for example: 42 = (10 – 6)2 32 = (7 – 4)2 52 = (8 – 3)2 Pupils should then be able to apply the idea confidently to such squares as

and

292 = (30 – 1)2 = 302 + 12 – (2 × 30 × 1) = 900 + 1 — 60 = 841 172 = (20 – 3)2 = 202 + 32 – (2 × 20 × 3) = 400 + 9 – 120 = 289

As a result of the foregoing exercises with squares, pupils are equipped, as for multiplication generally, with the widest variety of methods for finding squares. Thus: 282 = 28 × 28 = 42 × 72 = (25 + 3)2 = (30 — 2)2

246

Chapter 11 The Study Of Numbers Up To 1000—Ii

Some Special Exercises In The Manipulation Of Squares Gattegno is, however, not only concerned with reaching the goal of equipping his pupils with a simple means of finding squares up to 1000, but he also grasps the opportunity to give his pupils some extra games with squares to promote their intellectual interest and develop their general mathematical skill. These are clearly described in exercises 37–41 of Part IV of Book II of Mathematics. 1 The difference between two squares (for example, 92 — 42 = (9 + 4) × (9 — 4)) is studied by using the rods in such a way that the relationship is made very evident and understandable. 2 Some special differences between two squares are studied, such as when the results can themselves be expressed as squares; for example: 132 – 122 = (13 + 12) × (13 – 12) = 25 = 52 52 – 42 = (5 + 4) × (5 – 4) = 9 = 32 This discovery is followed by building of the three squares concerned and moving them into such a position as to form a triangle which is found to be right angled. 3

It is discovered that 1 + 3 = 22

247

The Cuisenaire Gattegno Method Of Teaching Mathematics

1 + 3 + 5 = 32 1 + 3 + 5 + 7 = 42 and so on, by means of a pyramid of rods adapted to enable the relationship to be seen. 4 It is similarly discovered that 2+4=3×2 2+4+6=4×3 2+4+6+8=5×4 and so on. 5 Finally, using these discoveries about the sum of the first odd numbers and the sum of the first even numbers, the pupils are guided into combining the two ideas, enabling the sum of the first whole numbers to be rapidly found: 1 + 2 = 12 + (2 × 1) 1 + 2 + 3 = 2 + (2 × 1) 1 + 2 + 3 + 4 = 22 + (3 × 2) and so on up to the first 20 whole numbers.

248

Chapter 11 The Study Of Numbers Up To 1000—Ii

The Reading And Writing Of Numbers It will be remembered that it was considered essential, throughout the course, for pupils to understand numeration beyond the particular range of numbers being studied. To conclude this chapter, a brief summary is given of the method of teaching the reading and writing of numbers beyond 1000–as recommended by Gattegno is Part VI of Book II of Mathematics. Pupils know how to read three-figure numbers but the signs they use are now set out vertically in three lines in the following manner: 1

2

10

20

100

200

3 30

4

5

6

40

50

60

300 400 500 600

7 70

8

9

80

90

700 800

900

By pointing to any two of the signs on different lines but pointing first to a sign on a lower line (e.g. first to 70 then to 3, or to 700 and then to 60, or to 500 and then to 4) the names of the numbers can be easily read. Similarly, any three signs on different lines can be pointed to starting with one on the bottom line, moving to one on the middle line, and then to one on the top line (e.g. 700, then 60, then 4 = 764). The only difficulty here is the irregularity of naming the numbers 11, 12, 13, . . ., 19, as eleven, twelve, thirteen . . ., and not as ‘ten one’, ‘ten two’, ‘ten three’, . . . For example, by pointing to 600, then to 10, then to 5 the name ‘six hundred ten five’ is produced which is called ‘six hundred and fifteen’.

249

The Cuisenaire Gattegno Method Of Teaching Mathematics

After vertically writing and reading numbers in the following manner: Top line

3

2

9

7

1

3

4

Middle line 7

3

6

7

2

5

7

Bottom line 6

4

8

7

1

5

6

4

9

and then modified slightly, thus: Top line

4

Middle line

2

3

5

Bottom line 6

5

8

8

7

the need for the zero as a ‘place-holder’ is realized when attempts are made at writing the numbers horizontally. Thus the second table becomes Top line

0

0

Middle line 0

3

4

Bottom line 6

2

0

4

0

5

8

5

7

9

0 8

providing, in their horizontal form, the writings which are already known. What is required now for giving the pupils knowledge of the reading and writing of numbers beyond 1000 is to introduce a succession of commas with different names according to where they are placed in the groups of figures.

250

Chapter 11 The Study Of Numbers Up To 1000—Ii

The first of these commas is called ‘thousand’, so that 2,783 is easily read as two thousand seven hundred and eighty-three. The pupils are then asked: ‘Can you read these numbers: 4,633; 15,217; 144,638?’ Note that the zero must still be used as a place holder in such numbers as 12,027; 102,003; and 12,000. If a second comma is introduced to the left of a six figure number, it is called ‘million’. Thus, the names of such numbers as 3,471,534 and 154,655,102 can now be read. Again, place-holders are used in such numbers as 73,001,040 and so on. The next comma on the left is called ‘thousand million’ so that now numbers of up to twelve figures can be read; for example, 3,215,474,689 and 708,643,294,062. By using other commas, read as ‘million million’ and ‘thousand million million’, any number that is not too long to write down can be read by simply using the names of the commas.

251

Chapter 12 Discovering Systematic Procedures

Care has been taken throughout these discussions not to confuse operations as mathematical concepts with the various procedures associated with them. This explains the relatively long period devoted to providing experience with the four operations before any one procedure was selected and adopted. Indeed it has been seen that the Cuisenaire-Gattegno course provides pupils with a broad background and a wide choice of methods. Through the Cuisenaire rods, pupils have been allowed to discover methods for themselves, the teacherâ€™s role being reduced to creating situations in which the need for them would arise. These methods may be regarded by some adults as rudimentary and inelegant, but, as in the case of child art, this is a feature of the arithmetic of children, particularly in the early stages when

Â

253

The Cuisenaire Gattegno Method Of Teaching Mathematics

small numbers are being studied. Furthermore, methods found and used by the child capture his interest and give him confidence—he knows that even when working mentally, the rods are always available for checking his own answers.

Horizontal And Vertical Notation As a result of this lack of formality, written work has been set out horizontally rather than vertically. This, as has been pointed out, is the language form of the sum, where the numbers are considered as wholes and not dissected into their hundreds, tens and units. As the realization of number value has been a feature of this teaching technique and as the numbers concerned have been comparatively small, the horizontal notation has been the more suitable up till now. It has also been more helpful than the vertical one in consolidating the notion of equivalence. However, in accordance with the principle of presenting situations in many different ways, vertical notation may be introduced as an alternative method of writing sums. It is recommended, however, that this be done only after the principles involved in a given situation are thoroughly understood by the children. It will be remembered that at all times throughout the course written expression of mathematical notions was introduced only after understanding had been ensured by oral means. Gattegno insists that ‘the child must know what he is doing before he is

254

Chapter 12 Discovering Systematic Procedures

expected to set down in notation what he has done’ (p. 21. Now Johnny Can Do Arithmetic). He continues, referring to the moment when the actual operation has been mastered: ‘Children do not mind what notation is used, and are quite ready to adopt another . . . It is just a matter of language and they are already used to hearing things in different ways; for example: “It’s time you were in bed”, or “Now you must run along to bed”, or “Off you go. You ought to be asleep”. It is the sense of what is said that matters, and variations in the words that give the sense are accepted without question’. Gattegno wrote the above whilst discussing the question of showing pupils 4) 13 as an alternative method of writing 13 ÷ 4. It is clear that if vertical notation is used exclusively, the teacher will find it much more difficult to express certain ideas, and may be tempted to introduce fixed procedures before pupils have understood the reasons for their existence. The reader is recommended to adopt the horizontal notation, introducing the vertical notation at a relatively late stage of the course, and then only as an alternative which condenses the wide variety of informal methods discovered; it should not be introduced until the pupils have mastered the operation concerned. Thus, in a second grade, addition might be shown both as 1

37 + 28 = 50 + 15 = 65

255

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

and as 2

or

As an understanding of vertical notation is an important step in the learning of any routine process, it would thus be best introduced just before the procedure is given in its more general form. This, of course, does not mean that the fixed processes in all operations should be introduced simultaneously or at any particular stage. Although the course is a single entity and continues without pause whether the numbers being studied are under 10, under 100, or under 1000, its various strands must at some point be brought together. Once pupils understand the operation concerned and have automatic response with the basic number facts, these strands should be tied together by means of a process that can be applied to all cases. It would appear that the processes for addition and subtraction can be introduced well ahead of those for multiplication and division. Teachers must decide this for themselves, but first pupils should be allowed to set out vertically the informal methods expressed previously in horizontal notation.

Â

256

Chapter 12 Discovering Systematic Procedures

Discovering The Fixed Procedure For Addition 1 The first step is to show pupils that a vertical notation is simply an alternative way of writing the horizontal notation. This may be introduced during the study of numbers up to 100, and the rods in a series of patterns can be used to show

This, of course, is equivalent to 49 + 27 + 18 = 70 + 24 = 94 This setting out can also be used for larger numbers

Â

257

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

When these methods have been introduced, pupils should be asked which is the easier and quicker method of setting out; thereafter they should generally be allowed to work either vertically or horizontally as they please. There should be no stipulation of any kind as to which column of figures, or even which combination of figures, should be added first. For example, taking 250 + 350 + 300 together, one pupil might write

Another, taking 250 + 350 first, could write

Â

258

Chapter 12 Discovering Systematic Procedures

In this way, the childâ€™s natural and free exploration of informal methods would not be discouraged. 2 The second step would be to introduce a fixed procedure by relating it to the pupilsâ€™ previous experiences. Even when pupils are studying numbers up to 1000 it is best to introduce the process with a group of numbers totaling less than 100. Working first with the rods, the pupils can be led to understand the principles involved in the procedure, enabling them at a later stage to work with great confidence on larger numbers. This can be shown by taking the sum 16 + 34 + 24 as an example. The three trains of rods representing these numbers are first made. (For greater clarity, the rods as shown here as separate entities, although they would of course, be placed end to end.)

Â

259

Â

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 85. The pupils will readily rearrange the rods in two groups, moving the orange rods to one side, since they already know how easy it is to add tens together. They then add the units. Since the result of this addition is expressed in the decimal system, that is by using as many orange rods as is possible and supplementing the length with the appropriate rod, they will in this case have formed the length orange plus pink. They can see that by putting the orange rod with the others, the answer immediately becomes apparent: 7 orange rods + 1 pink rod

Â

260

Chapter 12 Discovering Systematic Procedures

or in figures 70 + 4 = 74

FIG. 86. This knowledge, acquired with the rods, can easily be transferred to the vertical notation, since the same sequence of operations is involved: 1

Arranging the numbers vertically:

2 Adding the units:

Â

261

The Cuisenaire Gattegno Method Of Teaching Mathematics

3 The operation then becomes

However, if pupils understand die operation so well that they have no need for the rods, the process can be explained directly from the vertical setting out:

becomes

Once introduced, the idea of ‘carrying’ should be practiced often. If necessary, the association between rods and figures can be repeated using crossed rods for totals up to 1000. 3 An important feature of the Cuisenaire-Gattegno method, and one which has not so far been emphasized, is the checking of answers. The habit of using the rods to check answers that have been worked out mentally should always be encouraged.

262

Chapter 12 Discovering Systematic Procedures

The task of developing in pupils the ability to add numbers is now complete. They should be capable of using not only the methods they have discovered, but also of following the systematic process demanded by convention.

Discovering A Procedure For Subtraction Before the fixed processes for subtraction as recommended by Gattegno are described, the main points which have been developed in the earlier stages of this course concerning subtraction are summarized. 1

Subtraction finds the difference between numbers

In the initial stages of pre-number and of numbers up to 10, the aim was to develop the notion of subtraction as an operation which finds the difference between numbers (rather than as a process in which one number is ‘taken away’ from another). To achieve this aim, pupils progressed through three steps, each of which was largely a matter of language. Firstly, subtraction was closely related to addition by confining the activity to the words ‘what must be added to . . .’ . Thus, the simple example ‘what must be added to the length of the yellow rod to make it equal that of the blue rod?’ is extended to ‘what must be added to the lengths of the red and light green rods to make them equal that of the blue rod?’ or to ‘find two rods of

263

The Cuisenaire Gattegno Method Of Teaching Mathematics

different colors (or of the same color) which when added to the yellow rod make a length equal to that of the blue rod’. The term ‘difference’ was then introduced and experiences provided with ‘which rod makes up the difference between the yellow and the blue rods?’ This was simplified in time to ‘find the difference between 5 and 9’. The third step developed the ability to express subtractions in written form and involved the use of the sign for ‘minus’. Pupils should realize that the activities carried out in each of these three steps are the same, in other words, that ‘What must be added to 56 to make 72?’ is equivalent to ‘72 minus 56’. 2

The notion of equivalent differences

The second point developed in this course is the idea that subtractions form families, that there is an infinite number of subtractions that have the same difference; for instance: 5=5–0=6–1=7–2=8–3=... Experience of this fact is given because it forms the basis of the method to be employed in the operation of subtraction. It enables pupils to transform ‘hard’ subtractions into easy ones, to use 17–10 when faced with the equivalent difference of 16–9, to use 76–30 for 73–27, and 918–700 for 905–687. 3

Developing informal methods of subtraction

264

Chapter 12 Discovering Systematic Procedures

In the practical application of theory of equivalent differences as a means of carrying out subtractions, the pupil must either add a value to both the numbers concerned in the subtraction or subtract a value from both. For example, in a subtraction of the type 76–45, the difference is more easily obtained when 40 is subtracted from both numbers and 36–5 used as the equivalent difference. On the other hand, in the case of 76–48, in which ‘borrowing’ is normally used in traditional classes, the difference of 28 is easily obtained by adding 2 to each number and using the equivalent 78–50. If pupils can be taught to understand when it would be best to add and when to subtract a value from the numbers concerned, then, as Gattegno says, all difficulties connected with the operation will disappear. But is there an efficient method of leading pupils to understand and apply equivalent differences for carrying out the operation? First, rod patterns are arranged in such a way that the difference can be easily read without inserting rods in the space that represents the difference. How this might be attempted is briefly illustrated below. 1

Set out the rod pattern showing 17–5 or 17–8. Normally pupils will arrange their rods as illustrated, leaving the space showing the difference on the right.

265

The Cuisenaire Gattegno Method Of Teaching Mathematics

FIG. 87a. 17 – 5 =

FIG. 87b. 17 – 8=

2 Try and read the difference without using rods to fill the space. 3 See if there is an easier way to read the difference. (Pupils move the rod representing the second number to the right so that the space for the

difference is on the left.) FIG. 88a. 17 – 5 =

FIG. 88b. 17 – 8=

4 Now you can read the difference without using rods to fill the space. The second step is attempted when numbers up to 100 are being studied and subtractions involving larger numbers are given. 1

Arrange your rods to represent 56 – 35 (and 56 – 38) using rods end to end.

2 Subtract 30 from each number. difference remain the same?

Does

the

3 Try to read the difference without using rods to fill the space. (This may be found difficult by some.)

266

Chapter 12 Discovering Systematic Procedures

4 Move the rod representing the second number to the right so that the space for the difference is seen on the left. 5 Now try to read the difference without using rods to fill the space. The difference is easily seen to be 31 in the first example and fairly easily to be 28 in the second. It may be necessary for some pupils to use a rod for the units figure in the second instance (another tan rod in this case); use can be made of this difficulty to introduce the third step, which involves subtractions where the addition of a value to both numbers allows the difference to be easily read. 1

Make the pattern for 56 â€“ 38 again.

2 What must be added to 38 to make 40? 3 Add 2 to both numbers. 4 Can you read the difference quite easily now? This series of activities should be attempted by second grade pupils. Of course, finding the difference by means of the rods should precede exercises for which rods are not used. The exercises involving the larger numbers should later be carried out with crosses for the patterns instead of rods end to end. Finally, differences should be found without using any rods at all. When this stage is reached, the operation will have been thoroughly understood.

Â

267

The Cuisenaire Gattegno Method Of Teaching Mathematics

4 Vertical notation At a particular moment to be chosen by the teacher, pupils become familiar with the vertical notation for their subtractions. This may be done as late as when numbers up to 1000 are being studied. Thus 1

700 – 350 = 750 – 400 = 350 = 400 – 50 = 350

may be written:

2

712 – 389 = 713 – 390 = 723 – 400 = 323 or 712 – 389 = 412 – 89 = 423 – 100 = 323

can be written:

or

268

Chapter 12 Discovering Systematic Procedures

3

765 – 342 = 763 – 340 = 723 – 300 = 423 or 765 – 342 = 465 – 42 = 425 – 2 = 423

can be written

or

5

The final process

Pupils should now be ready to apply the notion of equivalent differences to the subtraction of such large numbers as 100,000,000 — 99,999,995 In this case the addition of 5 to each to obtain

269

The Cuisenaire Gattegno Method Of Teaching Mathematics

100,000,005 — 100,000,000 would easily give us 5 as the answer. More ‘complicated’ examples can be tried in which in the beginning one step transformations are made starting from the right; for example:

1

8 is added to each number (10,839 – 8,460);

2 40 is added to each number (10,879 – 8,500); 3 500 is subtracted from each number (10,379 – 8,000); 4 The difference, 2,379 can easily be written down. Similarly with 207,101 – 65,784, step by step transformations are again made but this time each number that is to be added will be noted down in turn:

The number in brackets is the complement of 65,784 in 100,000:

270

Chapter 12 Discovering Systematic Procedures

The operation could be made purely mechanical in the following way: Note that 6 + 4 = 10 while 8 + 1 = 7 + 2 = 5 + 4 = 6 + 3 = 9, so that the complement of the number to be subtracted to the nearest power of 10 only requires the knowledge of the complements in 10 and in 9. This then leads to the easiest of all transformations. Pupils are then challenged to write down any subtraction, changing it to one in which the second term becomes the appropriate power of ten, always remembering that they must keep to an equivalent difference. 6 Checking answers Pupils should be reminded of their earlier experiences concerning the relation between addition and subtraction. Knowing that if 7 – 3 = 4 then 4 + 3 = 7, they can check every subtraction by adding their answer to the number in the second term. For example:

271

The Cuisenaire Gattegno Method Of Teaching Mathematics

Discovering The Procedure For Multiplication First, the use of vertical notation as an alternative to the horizontal writing so far employed is adopted. If introduced in the early stages, it may be possible to show the easier and quicker ‘carrying’ method when calculating mentally. Some of the informal methods expressed in vertical notation are examined below: 1

Multiplying the units first:

2 Multiplying the tens first:

272

Chapter 12 Discovering Systematic Procedures

3 Saying (4 × 50) – (4 × 2):

4 Using ‘fours’: (18 × 14 = 3 × 6 × 2 × 7, and changing the order of the factors to 7 × 6 × 3 × 2 to give 42 × 3 × 2)

5 Using (3 × 250) + (3 × 6):

273

The Cuisenaire Gattegno Method Of Teaching Mathematics

6 Using the distribution of a sum over another (4 × 9) + (4 × 20) + (30 × 9) + (30 × 20):

From these examples it should be agreed that any of the multiplications set out horizontally can also be set out vertically. Pupils must first master this vertical notation before any attempt is made to introduce ‘carrying’. For this reason (and others) it would be advisable to postpone the introduction of the process until pupils are well advanced in the multiplication of numbers up to 1000. The steps to be taken then are listed below. 1

Demonstrating ‘carrying’ in association with the rods. Teachers will be able to devise a method similar to that adopted for addition. 274

Chapter 12 Discovering Systematic Procedures

2 Consolidating the ‘carrying’ process with multiplications of all kinds, at first with those for which pupils know alternative methods; for example:

with the answer to be checked both by dividing the answer by 5 or by an alternative method of multiplication: 5 × 76 =

× 760 = 380

3 Extending this technique to larger numbers:

4 Including in the above some examples of multiplication by multiples of ten (20, 30, 40 . . . ) and of hundred (200, 300, 400. . . ). 5 Multiplying by numbers of two or more figures first setting out each multiplication separately and totaling the separate answers; for example:

275

The Cuisenaire Gattegno Method

Of Teaching Mathematics

then showing how these can be done from the body of the sum and written ready for addition:

Operations on smaller numbers would be attempted first, particularly those for which pupils know alternative methods; for example:

can be checked by 25 × 27 =

276

× 2700 = 675

Chapter 12 Discovering Systematic Procedures

6 Casting out of 9’s. For multiplications involving larger numbers, pupils should be encouraged to apply a special check based on their earlier discovery of the test for divisibility of numbers by 9 Since this test is clearly explained in exercise 20, Part VII of Book II of Mathematics, the explanation given here is restricted to indicating how it would be applied to the multiplication 1,356 by 743, above. Cast out the 9’s from the sum of the digits in each number, thus:

Multiply the remainders obtained from the two numbers to be multiplied and again cast out the 9’s: 6 × 5 = 30

30 — 27 = 3

This remainder is seen to equal the remainder 3 found by casting out 9’s from the answer. When this equality is obtained the multiplication is probably correct.

Discovering The Procedure For Division 1

‘Milestones’

277

The Cuisenaire Gattegno Method Of Teaching Mathematics

In these chapters (as far as the operation of quotition division is concerned), the discovery of certain multiplication facts which can be used as ‘milestones’ in division has been emphasized. Thus, when the child knows that 8 × 23 = 184, he readily understands that 184 ÷ 23 = 8. He has also been discovering that his ‘milestones’ are valuable aids to the working out of divisions involving remainders. Thus, knowing that 8 × 23 = 184, he understands not only that 190 = 8 × 23 + 6 but also that 8 × 23 = 190 – 6 190 ÷ 23 = 8 with remainder 6 and

190 ÷ 8 = 23 with remainder 6

For an understanding of division, the value of basic multiplication facts in carrying out the operation must first be realized. When learning the procedure, pupils are asked to write down divisions in their multiplication form; that is, to discover the ‘milestones’ that will assist in each particular instance; for example: 1

86 ÷ 10 =

8 × 10 = 80 = 86 – 6

2 289 ÷ 17 =

172 = 289

3 569 ÷ 25 =

25 × 22 = 550 = 569 – 19

4 639 ÷ 32 =

20 × 32 = 640; hence 19 × 32 = 640 – 32; 19 × 32 = 608 = 639 – 31

278

Chapter 12 Discovering Systematic Procedures

5 897 ÷ 63 = 896

14 × 64= 896; hence 14 × 63 = – 14; 14 × 63 = 882 = 897 – 15

Secondly, these may be set out more simply in vertical notation;

2

Repeated subtraction’

It must further be realized by pupils that division means ‘repeated subtraction’. For instance, when asked to find ‘how many 4’s in 13’, pupils are given the opportunity to discover how many times 4 can be repeatedly subtracted from 13, and how much remains when these subtractions are completed. Their experience with the rods developed this notion of quotition division which now becomes the basis for Gattegno’s method of introducing and understanding the process that is commonly referred to as long division. Thus, in the last example above, the pupil who knows his ‘milestones’ may simply write

279

The Cuisenaire Gattegno Method Of Teaching Mathematics

For other pupils, it will involve finding out how many times 63 can be subtracted from 897. A few may have to subtract 63 repeatedly from 897, but others may use the fact that 2 × 63 = 126; or that 4 × 63= 252, or that 8 × 63 = 504. The paths followed by different pupils to arrive at the solution would therefore look like this :

280

Chapter 12 Discovering Systematic Procedures

It is in this way, says Gattegno, that the pupil learns that ‘divisions can be carried out in as many different ways as . . . subtractions can be repeated’ and, at the same time, each pupil can make use of his knowledge of ‘milestones’ to shorten the process. More difficult examples than the above can be seen in exercise 23, Part VII of Book II of Mathematics. Teachers should easily see that if they wish to conform to the usual procedure for division the same technique as above can be applied: 281

The Cuisenaire Gattegno Method Of Teaching Mathematics

It will also be realized that Gattegno’s method of carrying out the operation will eliminate, among other things, the ‘zero difficulty’, as in the following examples:

282

Chapter 12 Discovering Systematic Procedures

Â This concludes the Cuisenaire-Gattegno course in elementary arithmetic (except for the study of large numbers to be discussed in Volume II). First, informal activities with small numbers were studied and fundamental mathematical concepts of the four basic operations progressively developed; these were at last applied to the fixed procedures which are essential when operating on large numbers. In the next volume the use of the Cuisenaire materials for such special purposes as the teaching of fractions, decimals and percentages, English money and measures, problems and geometry will be discussed.

Â

283 281

Some Publications On The Cuisenaire Gattegno Method

Pupils’ Textbooks Mathematics with Numbers in Color

by C. GATTEGNO

Book I

—Qualitative arithmetic The study of numbers from 1 to 20

Book II

—Study of numbers up to a thousand The four operations

Books III —VII continue the course through to the Secondary School level. Teachers’ Manuals Now Johnny Can Do Arithmetic

by C. GATTEGNO

A guide for parents and teachers on how the rods are to be used. A Teachers’ Introduction

by C. GATTEGNO

To the Cuisenaire-Gattegno method of teaching arithmetic. Talks for Primary School Teachers

by M. GOUTARD

A practicing teacher advises her colleagues on Cuisenaire in the classroom.

285

The Cuisenaire Gattegno method of teaching Mathematics

Published on Nov 18, 2009

The Cuisenaire Gattegno method of teaching Mathematics

Advertisement