Nuffield Maths Teacher's Handbook 1

Nuffield Moths 5-11a revised and extended version ofthe Nuffield Mathematics Teaching Project

Nuffield Moths 1Teachers' Handbook

11~~~!llrI~I~IIIII~IIIIN29720

Published for the Nuffield Foundation by Longman Group Ltd

II

General editor:Eric A. Albany,Senior lecturer in mathematicsThe Polytechnic, Wolverhampton

Author of this volume:Winifred Moore,Headmistress ~

NATrtmfMw~1First School,~!JljUi ~,,,..

1...~.~.fi,;~~:,:::&~t.',.h?r,.."s..,.:-(,',:..:!JJ'.:.~.tJ:o:~meriy,Headml~tr~,s ,i,~tMornt1[~~;;!!1JaJ!Jf'Scho~~London

C .~-eo CJJfd Do-.dzrector of the 'E NEJzrR~thematical Experiences ProjectEric A.f\lbany

Illustrator:Chris Williamson

We are grateful to Mrs. S. Daddand the children of Nelson Infants' School,Napier Road, London E. 6for the photographs usedthroughout this book.We are also grateful toPhilip & Tacey for the use ofthe rubber stamps which arepart of some of the illustrations.

LONGMAN GROUP LIMITED

LondonAssociated companies, branches and representativesthroughout the world

The Nuffield Foundation 1979All rights reserved. No part of this publicationmay be reproduced, stored in a retrieval system,or transmitted in any form or by any means, electronic,mechanical, photocopying; recording , or otherzoise,without the prior permission of the Copyright owner.

First published 1979Second impression 1979ISBN 0582 188849

Photo set in Monophoto Plantin 110 and 194by Keyspools Ltd., Golborne

Printed in Englandby MCCorquodale (Newton) Ltd.,Newton-Ie-Willows, Lancashire.

ContentsForeword by Geoffrey MatthewsIntroduction

V

VI

t Sets and relations (Nt)

I I I I I I I I I ! I I I I

Nt: 1 Relations (different types of correspondence) 2Nt:2 Early sorting experiences -leading to sets 6Nt:3 Sorting into subsets 9

2 Matching (N2) .14

tl I I I I I I II

I

I I I

N2: 1 Matching to find equivalentsets 15N2: 2a Two non-equivalent sets 20N2: 2b Three non-equivalent sets 21N2:3 Pictorial representation and semi-tallying 23

3 Counting and rrurrrer-als (N3) 26

I

I ! II I I I I

iI

I I I

N3: 1 Counting 27N3:2 Matching a number symbol to a set 28N3:3 Introducing number words 34N3:4 The empty set 37N3:5 Conservation of number 38

4 Ordering (N4) 42

I I I I

I

II

I I I I I I I

N4: 1 Ordinal numbers 43N4:2 Putting non-equivalent sets in order 47

I

N4:3 Signs 'is greater than', 'is less than' 50N4:4 Tallying and pictorial representation 53

5 Towards addition (N5) 56

N5: 1 Composition of small numbers 57la bead bag 57Ib duck pond 58lc rods or strips and frames 60

I I I II

I I I I I I I I

N5: 2a Addition of two disjoint sets; putting sets together 63N5: 2b Addition of two disjoint sets, using structured apparatus 64N5:3 Recording addition by mapping 65

6 Addition to to (N6) 68

N6: 1 Number bonds up to 10 69N6:2 Counting on 75N6:3 Patterns in simple addition 80N6:4 Picture problems - additions 82N6:5 Introduction of addition sign and vertical addition 83N6:6 The addition square 85

111

7 Length (Ll) 88

I I I I I I I I I I I I I I

Ll: 1 Descriptive language 89Ll: 2 Comparing two unequal lengths 92Ll: 3a Matching lengths: matching two objects of about the

same length 95Ll:3b Matching lengths: using several objects to 'make up' a

length 96

I I I I I I I I I I I I I

Ll: 4 Ordering 97Ll: 5 Measuring with repeated units 99Ll:6 Using limb measures 101

8 Shape and space (SI) 103


SI: 1 Awareness of shape and space, extension of vocabulary 105S1: 2 Early sorting of 3-D shapes 109SI: 3 Early sorting of 2-D shapes 111Appendix 115

9 Weighing (WI) 117


WI: 1 Descriptive language - heavy and light 118Wl:2 Comparing - heavier than and lighter than 119Wl:3 Balancing 123Wl:4 Ordering 127

10 T'Irrre (Tl) 131


Tl: 1 Association - matching events to daytime or night-time 132Tl: 2 Putting time sequence in order 134Tl: 3a Comparisons: fast and slow 135Tl: 3b Comparisons: timing 136Tl: 4 Graphs and charts 137

11 Money (Ml) 141

I I I I I I I I II I I I IMl: 1 Recognition of 1p, 2p, 5p, lOp coins 142Ml:2 Comparison of amounts of money

(a) by matching (b) by totals 143


Ml:3 Using coins to make amounts up to lOp 144Ml:4 Early stages of shopping 146Appendix: Suggestions for shops in schools 149

12 Capacity (Cl) 153

Cl: 1 Full, empty and half empty 154Cl: 2 Which holds more? (visual judgement) 155Cl: 3 Finding the capacity by counting 156Cl: 4 Sorting containers 157Cl: Sa Comparing by emptying 158Cl: 5b Comparing by filling 158

IV

ForewordAs organizer of 'N uffield Mark 1', I am delighted to have the opportunityof welcoming the present publication, which in effect is Nuffield Mark II.The original project started in 1964 with the aim of 'producing acontemporary course', an urgent need at the time when the II-plusexamination in arithmetic was on its way out and there was a realizationthat neither its contents nor the methods of teaching for it were producinghappy or numerate children (the overwhelming majority of people in thatera grew up to hate and fear the subject).The decision was taken at that time to produce only guidance for the

teachers of primary children and not materials for the children themselves.Arguments will continue to rage as to whether this was a wise decision. Ican defend it vigorously on many counts at that time, but I am also glad tobe on record as saying that about ten years later there would be theacceptance and the need for the production of pupils' materials as well.And so, of course, it has turned out. Very many teachers have asked for

more guidance and more materials to be put into the hands of their pupils,and this is just what Mark II has set out to achieve. It is very fortunate thatthis enterprise has been directed by Eric Albany. He is a staunch Nuffieldman who contributed a lot to Mark 1. His ability, shrewdness and sense ofboth humour and proportion have ensured that Mark II should completethe task of 'producing a contemporary course' which children can enjoy. Ofequal importance, they will be helped to think for themselves and toacquire relevant skills to the very best of their ability. Eric Albany and histeam have produced materials which will set a standard of excellence formany years to come.Among the many institutions and people to whom the Foundation owes

thanks for their help, I must especially acknowledge the part played by thePolytechnic, Wolverhampton in allowing the full-time secondment ofEric Albany to the project and also the assistance given by Wolverhamptonand Walsall Education Committees in providing accommodation andfacilities for the project staff. We are extremely grateful to all those teachersand schools who have taken part in the trials of the new materials. I wouldalso like to express our thanks to William Anderson, Publications Managerof the project and his colleagues, to the project secretary, Kathleen Norton,and to our publishers, Longman Group Ltd, who have devoted so mucheffort and such skill to the editing, design and production of the materials.

Geoffrey Matthews

Chairman of the Nuffield FoundationPrimary Mathematics Consultative CommitteeProfessor Emeritus, Chelsea College,University of London

v

IntroductionNuffield Maths 5-11 is based on the original Nuffield Mathematics TeachingProject but is revised in the light of experience, and extended to include thefull range of pupils' materials.The materials for rising 5 to 7 year olds are:

Bronto Books, Sets A, B, C and D-colourful books linking the extensionof mathematical vocabulary with language development (4t-6 year olds);Nuffield Maths 1 Teachers' Handbook and expendable worksheets (4t-6year olds);Nuffield Maths 2 Teachers' Handbook and expendable worksheets (5t-7year olds)

Teachers' Handbooks 3, 4, 5, 6 and non-expendable pupils' books are suitablefor 7-11 year olds.

Aims and objectives

The general aim of the Nuffield Maths 5-11 Project is to promoteunderstanding of the concepts and proficiency in the basic skills ofmathematics in children of the 5-11 age range.The objectives of the Teachers' Handbooks for Nuffield Maths 1 and 2 are:a) To give teachers clear guidance on the content, methods and timing

appropriate at each stage of the course;b) To give practical, 'down to earth' suggestions for teaching Number,

Measurement and Shape, using activities suitable for children with awide range of abilities and backgrounds;

c) To give ideas for making worksheets, workcards, charts, models, etc.and guidance in the use of both homemade and commercially availableapparatus;

d) To stress the importance of linking the extension of mathematicalvocabulary with language development;

e) To suggest ways of dealing with children's difficulties.

Using the materialsThe materials of the Nuffield Maths 5-11 Project can be used in a variety ofclassroom organizations including individual work, group or class teaching.This should prove particularly useful to the teacher who tends to vary thetype of organization to suit particular topics. Whichever system is used, itis important for teachers to remember the following points:a) Children learn at different rates and so will not reach the same stage

simultaneously;b) Young children learn by doing and by discussion;c) As well as finding out and 'discovering' things about mathematics,

children need to be told things about mathematics, particularly if newvocabulary is involved.The obvious line of development for a primary child learning

VI

mathematics would seem to be:

activity and ~ thinking and f------ acquisition of skillsexperimentation communication and reinforcement

Activity and experimentation may vary from a child 'finding out by fiddling'to a structured or teacher-led activity.

Thinking and communication involves discussion, sometimes betweenchildren, sometimes between teacher and children. In the latter case, theteacher may pose a friendly 'I wonder' type of question to find out what thechild is thinking-or indeed if the child is thinking at all. Talking leads tosome sort of recording, the first introduction to which may be merely aplacing or arranging of objects on a prepared sheet of card. Drawing andsimple writing follow, leading to 'shorthand' and eventually to the use ofsymbols.

Acquisition of skills and reinforcement Apart from the obvious benefits ofhaving certain useful skills and facts at one's fingertips, there is thequestion of building up confidence and enjoyment 'I can do these, Miss.Can I have some more ?'The important thing is that these three elements form a sequence.

Worksheets provided by the Project are seen as part of the last element ofthe sequence. These expendable worksheets are printed in 4-page leafletform, with space for the date and pupils name. They should not be given tothe children before they have had ample opportunity for activity,experimentation, thinking and discussion.

Chapters and stagesThis Handbook contains: six chapters dealing with Number (labelledNl-N6); one chapter on Shape and Space (S 1); one chapter on Money(Ml), and four chapter on Measurement: Length (Ll), Time (Tl),Capacity (Cl), and Weighing (WI). Further chapters on these topicsappear in Nuffield Maths 2 Teachers' Handbook.

Each chapter is divided into stages which are also numbered. The secondchapter on Number, for example, contains three stages which are labelledN2: 1, N2 :2, N2 :3. (See Contents for details of stages in each chapter.)The chapters on Shape and Space, Measurement and Money need not

be introduced in the order in which they appear in this Handbook. Indeed,the early stages dealing with descriptive language in the Measurement andShape and Space chapters are intended to run concurrently with the earlywork on Number. Similarly, the stages devoted to sorting, matching,comparison and ordering of Length and Weight, etc. should be 'slotted in'at the appropriate points. A 'suggested order of development' is given onpage ix .

Vll

Next to the full list of Contents on pages iii+iv, there is a grid whichteachers may wish to use as a record of the stages reached by groups orindividuals.

Chapter formatEach chapter of the Teachers' Handbooks will be set out as follows:

1For the teacher:A brief outline of what is being attempted in the chapter, where it isleading, and what should have been covered before starting.

2 Summary of the stages:Setting out the stages contained within a chapter.

3 VocabularyA list of the words and phrases which the children will need to be able touse and understand if they are to appreciate and explore the ideas in eachchapter. The teacher may wish to include some of these words and phrasesin her work on language.

4 Equipment and apparatusThe sort of materials such as boxes, containers, pots, sticks, pictures,sorting toys, buttons, counters, cubes, beads, string, sand, plasticine, etc.which the teacher may need to collect in advance.

5Working with the childrenSuggestions for introducing and developing each stage through discussion,teacher-led activities, games, etc.; hints for making workcards, charts,models, displays and simple apparatus; how to check-up" where necessary,that a child understands a particular stage.

The accompanying worksheets, published by the Project, arereproduced on a reduced scale at the end of each stage after the reminder'Worksheets should not be introduced until children have had plenty ofpractical experience and opportunity for discussion.'

6 References and resourcesA list of books and commercially produced materials which are appropriatefor the chapter. No commercially produced equipment is deemed essentialbut is suggested as a possible alternative to homemade or environmentalr.zaterials. Occasionally the Teachers' Guides published by the originalNuffield Foundation Mathematics Teaching Project may be listed in thissection. These guides are now out of print but may be found in librariesand schools and they still make a valuable contribution to mathematicseducation.

V111

A suggested order of development

Nutnber Quantities and Shape

NI: 1NI: 2NI: 3

RelationsSorting - leading to setsSorting into subsets

LI: 1SI: 1WI: 1TI: 1LI: 2WI:2

Descriptive languageAwareness of Shape and SpaceDescriptive language - heavy and lightMatching events to day-time/night-timeComparing two unequal lengthsComparing - heavier than/lighter than

N2: 1 Matching to find equivalent sets

N2: 2a Two non-equivalent sets2b Three non-equivalent sets

N2: 3 Pictorial representation and semi-tallying

SI: 2SI: 3CI: 1TI: 2

Early sorting of 3-D shapesEarly sorting of 2-D shapesFull, empty and half emptyPutting time sequence in order

N3: 1N3:2N3:3N3:4N3: 5

CountingMatching a number symbol to a setIntroduction number wordsThe empty setConservation of number

L I : 3 Matching lengthsMI: 1 Recognition of l p, 2p, 5p, lOp coins.

N4: 1N4:2N4:3_N4: 4

Ordinal numbersPutting non-equivalent sets in orderSigns 'is greater than', 'is less than'Tallying and pictorial representation

L I : 4 Ordering length

CI: 2 Which holds more? (visual judgement)CI: 3 Finding the capacity by countingCI: 4 Sorting containers

N5: 1 Composition of small numbersIa bead bagIb duck pond1c rods or strips and frames

N5: 2 Addition of two disjoint sets2a Putting sets together2b Using structured apparatus

N5: 3 Recording addition by mapping

MI: 2 Comparison of amounts of money(a) by matching(b) by totals

LI: 5 Measuring with repeated unitsTI: 3a Comparisons: fast and slow

3b Comparisons: timing

N6: 1N6:2N6:3

N6:4N6: 5N6: 6

Number bonds up to 10Counting onPatterns in simple addition

WI: 3 BalancingCI: 5a Comparing by emptyingCI: 5b Comparing by fillingMI: 3 Using coins to make amounts up lOp.MI : 4 Early stages of shoppingWI: 4 Ordering weightLI : 6 U sing limb measuresTI : 4 Graphs and charts

Picture problemsIntroduction of' + ' signThe addition square

IX

Chapter 1

Sets andrelations (N 1)

For the teacherThe whole of mathematics has been described as sets and relations. Thisstatement is particularly relevant to young children, for they are probablyfamiliar with sets of objects, a tea set, a train set, a set of pencils, etc., andsoon develop the idea of a set as 'a collection of things which go together' .Relations (or relationships) arise as a result of a connection between twoobjects or two sets of objects. When a child says, 'That coat belongs to me,'or, 'There are more straws than milk bottles,' these are examples ofrelations.Much later on, we shall use these same ideas when dealing with numbers.

For example:

The set of even numbers (2, 4, 6, 8, ... )The set of square numbers (1,4,9, 16,25,36, ... )2 is less than 5. 4 is half of 8.

As always, it is advisable to begin with the child's present experiences andtry to extend them in the direction we want to take. Of particularimportance is the language the child uses to express all that he finds outfrom his new experiences. The language of a pre-school child is oftenrichly imaginative in its use, but we must also extend this to include notonly the names of the new materials and activities, but also the vocabularyof number words.

Summary of the stagesNt: 1 Relations (different types of correspondence)

Nt: 2 Sorting-leading to sets

Nt: 3 Sorting into subsets

1

Sets and relations (N!)

VocabularyBelongs to, lives in, has the same colour as, sits next to, match, matching,find its partner, colour, red, blue, yellow, green, silver, shiny, pretty, little,big, set, sets of, make a set of, subset, straight, curved, plastic, glass,rubber, wood, cork, sponge, feather, cotton, straw.

The Bronto Books produced by the Project introduce much of thevocabulary applicable to this stage.

Equipment and apparatusIt is suggested that the following are collected:

A piece of hardboard covered with flannel, large flat shirt box, a box of'pairs' -gloves, shoes, socks, mittens, slippers; box of 'identical pairs'-2red beads, 2 green square beads, 2 pairs of scissors, 2 pieces of blue Lego, 2white plastic pegs, 2 twopenny bronze coins, 2 pieces of chalk, 2 identicalrubbers.

1 red bead1 green plastic toy1 reel of white cotton1 blue square1 silver key1yellow piece of material

1 red piece of Lego1 green peg1white brick1blue pencilI silver button1yellow biro

Also collect 4 cut-out pictures to represent father, mother, son and baby,and sets of possessions belonging to each. 4 shoe boxes. The cut-outpictures should be backed with card and covered with Contact.

Cut out, back and cover pictures of things that swim, things that fly,things I eat, things I wear, farmyard animals, wild animals, and a largepicture of a house with cut out pictures of furniture to fit in each room.

Collect sets of objects displaying properties of: colour, shape, size,roughness, smoothness, soft, hard, holes, no holes, straight sides, curvedsides, heavy, light, will stand up, will not stand up, plastic, glass, rubber,wood, cork, sponge, feather, cotton, straw.

Working with the childrenNt: 1 RelationsA relation (which some might prefer to call a relationship) is a connectionor correspondence between the members of one set and the members ofanother.By noticing and talking about relationships which are well within a

child's understanding, his powers of observation are improved. (My coat isbrown, Mary's coat is blue.) These experiences, whether incidental orstructured, form an important part of the development of both languageand mathematics.

The teacher needs to think about the relationships the child is toencounter and provide situations in which they may be found. At the sametime, she must help the child to describe the relationships; graduallyworking towards a more sophisticated structure of language. Recording,

2

Sets and relations (N 1)

Todaywill giveout

8 G~Amanda~ ~

when appropriate, should be done with the help of the teacher, in the firstinstance. The arrow, with the relation written above it, is a usefulshorthand symbol which often reduces the amount of writing required.

('At the top it tells you what to say as you slide along the arrow. ')Name cards, which can be changed daily, slot into this more permanentdisplay. Children should always be encouraged to 'read' the recordedrelationship: 'John will give out the milk.' 'Amanda will give out thestraws.'

Relations involving one-to-one correspondenceIn the first instance, children need lots of experience

. with the relation between single objects only:

as a little lamb.

has

Board covered in lint or flannel

Later, a more complex situation might involvemore objects or people:

belongs to

~Peter

~ohn

~Sue

The bat belongs to Peter.The ball belongs to John.The yo-yo belongs to Sue.

The relation is between toys and children. There is a common relationshipbetween the bat and Peter, between the ball and John, and between the yo-yo and Sue.

Characters from well known nursery stories can be drawn or cut fromold books, stuck on cards, and faced with clear Contact. Each card can bebacked with flannel or a small round piece of plastic foam stuck on theback, making the picture adhere. In this way the apparatus can be keptclean, and used over and over again in different ways. A piece of hardboardwith flannel stuck on the front, and supports nailed to the back, makes anideal freestanding display. Another good idea is to cover the inside of alarge shallow flat box (similar to a shirt box) with flannel. This props upnicely and all the apparatus used with it can be kept tidily in the box whennot in use.

3

Sets and relations (Nl )

The first little pigbuilt his house of straw.

The second little pigbuilt his house of sticks.

The third little pigbuilt his house of bricks.

belongs to)

4

A large fiat tidy box, the inside covered with felt, can be used to illustratestories, such as The three little pigs:

Further examples are:

pumpkin, mice, rats 'turned into' (Cinderella).

one 'for' the master, etc. (Ba Ba Black Sheep).

In these cases there is a correspondence between one object or person onthe left of the diagram and one object or person on the right of the diagram.This early experience of matching to show one-to-one correspondence in arelation is very important and will be used, in due course, as a prerequisiteto counting.

Relations involving many-to-one correspondence Children who are able tohandle one-to-one correspondence with confidence may be ready to discussand record relations involving many-to-one correspondence:

was born inAnneLynneIanKaV'en -----~Colin

May

~ut1e

Notice that only one arrow is drawn from each member of the left handlist. Examples should be chosen so that each element on the left can berelated to one, and only one, element on the right. (In this example, eachchild can only be born in one month.)

A simple piece of apparatus illustrating many-to-one correspondencecan be made by pasting pictures cut from magazines or old books on to asheet of card. The card is then covered with clear Contact or stapled insidea clear polythene bag. The child records by using a felt tip pen to draw thearrows. These arrows can be easily removed by using a damp cloth so thatthe card may be used by another child.

Other examples might be: children who stay to dinners, others who go

is the partner of

eats

Matching to show a common relationship

Sets and relations eNI)

home; children wearing tights, long socks, short socks; things that float inwater, things that sink. There is an obvious connection between this kindof relation and sorting.

It is also possible to have a 'many-to-many' correspondence, for example:

likesAnn Chocolates

Beryl

Cleo Biscuits

On the whole, the one-to-one correspondence and many-to-one cases aremore interesting mathematically.

Special kinds of relations such as order relations ('is longer than', etc.)and equivalence relations ('is as many as') are considered in the chapters onlength, capacity, etc., and in the later chapters on number.

Early matching activities involving one-to-one correspondence

1 (Partners' Pairs of gloves, mittens, shoes, socks and slippers, etc. are splitup so that one of each pair is in one box or pile and its partner is in theother. One article is taken out and the children invited to 'find its partner'or 'find the one that matches this one'. 'Are you sure they match ?'Recording the relation 'is the partner of' needs a double-headed arrow~ because this relation may be 'read' in either direction.

2 (Twins' A collection is made of pairs of familiar objects which areidentical in every respect (colour, shape, size): two round red beads, twogreen square beads, two pieces of blue Lego, two white plastic pegs, twotwopenny bronze coins, two whole pieces of white chalk, two identicalrubbers, etc. The teacher takes one article and asks the child to find onethat matches it, or one that looks just like it. Record using the relation'looks the same as' and a double-headed arrow ~ .

3 Matching for colour, shape, etc. Prepare a box of pairs of objects whichcan be matched for colour only, for example:

One round red bead to match one red piece of Lego.One blue square bead to match one blue pencil.One green plastic toy to match one green peg.One silver key to match one silver button.One reel of white cotton to match one white Unifix brick.One yellow piece of material to match one yellow biro.

The teacher uses a similar instruction to the previous one, inviting thechild to find an object which 'has the same colour' or 'matches the colourof'. Plastic or card shapes are used in a similar manner for the relation 'hasthe same shape as'. Double-headed arrows are required when recording.

4 Matching to show a common relationship After discussing what pets andother animals eat, a chart could be made showing the common relationshipbetween each animal and its food. Notice that here a single-headed arrow isused because the relation 'eats' can only be read in one direction.

5


~ Number 1~ sheet 1.1

Here is a small boll. Here is a big boll.GJj:~~"bull ~.ive pe: th:e~:~~ boll ..~~~Shanti has a .boll. Peter has a boll.

Put a long tail on the mouse. Put a short tail on the dog.

The mouse has a toil. The dog has a

Match one-to-one.

~

~~qo i)

Match one-to-one.

4

6

tail.

sheet 1.4

On the worksheets the second column is rearranged so that the childrenhave to sele~t the correct 'partner' before joining ~ith an arrow.

Worksheets should not be introduced until children have had plenty ofpractical experience and opportunity for discussion

sheet 1.2Draw a toy for Shanti.

Draw a toy for Peter.

Draw a bone for Patch.

2

Number 1sheet 1.5

the sea

5

sheet 1.3

Match onf}e~. ~ __ ;,_the_po_,,"_.,o_r _~ ~

:$ ~L ~

3

sheet 1.6

~

~

. ,,

~

@>o

~

6

Nl : 2 Early sorting experiences (leading to sets)Children enjoy rummaging through mummy's button box looking for 'theshiny buttons', or 'all the pretty ones', or 'the little tiny ones'. Usuallythese 'sets' are played with and then put back in the box, and further setsselected according to new properties chosen by the child himself. Theproperty selected does not matter since the idea of a set is 'a collection ofthings which go together'. Children see sets of many kinds all around them,both at home and at school. Every opportunity should be taken to start adiscussion about sets of objects, for example, in the home corner there maybe a set of pans, a set of cutlery, a tea set. In the cupboard we might see aLego set (all different pieces for building), a train set (the lines, signals,

Sets and relations (NO

wagons, etc.), a Meccano set (the struts, nuts and bolts). The child maypossess a set of crayons, a set of coloured pencils, or a pencil set. It isadvantageous at this point to encourage the children to use the word 'set'when referring to a collection of articles which go together in some way.From this first play stage emerges the next, where the child will be asked

to consider objects as members ofa set, their relationship with each other,and with the set as a whole. In this way sorting helps to develop logicalthought and decision making.

Examples of sorting activities using pictures or objects Cut out pictures torepresent mother, father, son and baby daughter. Back each picture withcard and cover with Contact for durability. Make cut-out pictures of sets ofclothes and possessions which clearly belong to each person. Using foursmall shoe boxes, stand a figure at one end of each box and ask the childrento sort by placing each of the picture cards in the appropriate box, e.g. therattle belongs to baby, the pipe to father, the dress to mother, etc.

Pictures similar to those used for the example of many-to-onecorrespondence, can be sorted into sets enclosed in loops of wool or chalklines:

These thingsbelong to=

7


8

Pictures can also be sorted using a piece of manilla card andappropriately illustrated labels. The child selects a label and sorts throughthe pictures to find those which should be placed on the card with the label.

Other labels Things ~Iwear --uThitTg5t~t .c:fly ~

Wild ~anima(s~

Farmyardanimals

The teacher should check the child's sorting and discuss anyuncertainties. It is important that the child is able to justify his sorting, forexample, 'I've put the tiger there because it belongs to the set of wildanimals.'Assignments may be given verbally to a small group of children asking

them to 'make a set of cars', 'make a set of animals', 'make a set of boats' ,'make a set of red beads', 'make a set of small circles', etc. The objects maybe enclosed on trays, paper plates, tin lids or the teacher may wish toprepare a piece of manilla paper labelled thus:

IMake a set af CIIThe teacher places an object in theframe and with the children readsthe instruction.

Make a set of I LJl~ I

Should the teacher wish the child to keep a permanent record of hiswork, then the instruction may be copied into the child's book. The childcan then record by drawing the objects.

A set of flat plastic toys, which are easily drawn round, are ideal for thiskind of sorting.Another way might be to let the child cut out pictures from magazines

and catalogues and stick these directly into the child's book or on a friezefor display.

Later, more able children may wish to draw or write the names of objectsin a set for themselves.


Worksheets should not be introduced until children have had plenty ofpractical experience and opportunity for discussion.

Number 1sheet 2.1

sheet 2.2

2

Nt: 3 Sorting into subsetsSo far, sorting has consisted mainly of making a selection of 'things that gotogether'. Having obtained a set whose members all have something incommon, the next stage concentrates on the differences which may existbetween the members within the set. The original set is split or partitionedinto subsets; each object belonging to just one of the subsets.The set of toys has been partitioned into two subsets: one of dolls, one of

cars.

Children need lots of experience in partitioning. It is helpful to provideprestructured sets of objects displaying an obvious difference which willenable the objects to be partitioned into two subsets in the first instance.Further examples are:

a set of bricks-red and blue,a set of toys-cars and boats,a set of bricks-those that roll and those that do not roll,a set of shapes (same colour)-squares and triangles,a set of children-boys and girls.

Pictures pasted on card and covered in clear Contact can be used repeatedlyif children draw the rings with a felt pen.

Put aS round the things that betong together.

~

'" :'\ "1\"" .,,-... ..

I

,)

.

9


It is not necessary to record every activity, but paper plates or trays areideal for holding subsets of objects. The teacher can check these easily andprovide appropriate vocabulary.

On occasions the child may like to draw a picture of his sortingand label it.

Other properties which may be considered for sorting are:

Large, small; heavy, light (differences must be obvious); rough, smooth;hard, soft; wet, dry; with holes, without holes; will stand up, will notstand up; words beginning with ch, words beginning with sh.

~Gradually, sorting involving more than two subsets may be introduced.A large picture of the inside of a house showing four rooms can be'furnished' by placing cut-out pictures of furniture which obviously belongto a particular room.

The k itc.hen

Further examples of sorting into more than two subsets:

toys: dolls, animals, cars, boats.children: stay to school dinner, bring sandwiches,

go home.pets: dogs, cats, hamsters, etc.children: wearing long socks, short socks, tights.materials: made of wood, plastic, rubber, metal, etc.names of objects: beginning with b, d and p.

The bath r-oom

Ways of recordingLoops of string or coloured wool.Coloured plastic hoops or P.E. hoops.Paper plates.Chalk marks on the floor.Sticks or strips of card used to partition the set:

10

Sets and relations (Nl )

Trees or 'railway sidings' -each member is moved along thetrack into its correct subset.

A more permanent record can be made by using a chartshowing many-to-one correspondence.

is made of

mallet

~key~ plastic~ shape

~

plastio

screw ---> metalcar-~

This method of recording has been criticized becausethe diagram can become so cluttered with arrows that itlooks like a scene from the Bayeaux Tapestryor 'Custer's last stand'.

I is made of I'to .tO~~ ~@~

0


Children should be encouraged to talk about the way in which they havesorted, explaining the differences and similarities and suggesting thecorrect labels for the subsets. Sometimes it is a good idea for the teacher tostart the sorting and then ask, 'How am I sorting these?' 'What labelsshould we put on the subsets ?'

In some worksheets the child is given the chance to decide for himselfhow to sort. There is no single 'right answer', but the child should be ableto justify the way he has sorted.


~ Number 1~ sheet 3.1

Put aBround the things that beLong together.

e> ~ ~ ~0' 0> (j> b~'b~

Put aSround the things that belong together.

3

12

sheet 3.3

Put as round the things that belong together.

ODDD

2

Put a8round the things that belong together.

4

sheet 3.2

sheet 3.4


References and resourcesDienes, Z. and Holt, M. Let's Play Maths., Penguin 1973 (Sections on

sorting games and ping-pong puzzles contain many excellent ideas foruse with children from 4 to 5+years old.)

Matthews, G. and J. Toward Number, Early Mathematical Experiences,Addison Wesley 1978

Nuffield Mathematics Teaching Project, Mathematics Begins CD, NuffieldGuide, Chambers/Murray 1967 (See Introduction, page viii).

Grouping and Setting Shapes, Invicta Plastics

Grouping Sets, Metric Aids

House Sorts, Sea Sorts, Tree Sorts, E.S.A.

Plastic Sorting Toys, E. J. Arnold

Rubber Stamps, Philip & Tacey

13

Chapter 2

Matching (N2)

14

For the teacherWhen dealing with relations in Chapter 1, the children found 'partners' or'twins' by matching one-to-one. In this chapter we extend this idea tocompare the number of objects in sets. Gradually, the emphasis is changedfrom the appearance of the individual objects to the number of objectswithin a set. Instead of looking at the properties or characteristics of theobjects, we shall be concentrating on a particular property of the set. Thisparticular property of a set could be called its 'how-many-ness' or itscardinality.

By matching objects one-to-one we can compare sets to see if theycontain the same number of members or if one set has more members thananother. By matching one-to-one we are able to make this comparisonwithout being able to count.

Surrrmary of the stagesN2: 1 Matching to find equivalent sets

N2: 2a Two non-equivalent sets

N2: 2b Three non-equivalent sets

N2: 3 Pictorial representation and semi-tallying

Vocabulary

Match, matching, enough, not enough, as many as, one each, some leftover, more, less, fewer, more than, less than, fewer than, too many, too few.

Some people prefer to use 'less' when discussing substance and 'fewer'when discussing separate, countable objects, e.g. 'I prefer less sugar in my

Matching (N2)

tea so I will take fewer lumps.' The opposite of 'too many' is generallyaccepted to be 'too few' rather than 'too less'. However, at this stage, it ishardly worth being too pedantic especially as common usage may varyfrom region to region.

Equrprnent and apparatusPictures of cars, eggs, etc., mounted 011 card, square beads and string,coloured pegs and pegboard, shapes, sorting toys, bricks, cubes, etc.,clothes pegs, Poppet beads, interlocking cubes, curtain rings, washers orcotton reels for tallying.

Working with the childrenN2: 1 Matching to find equivalent setsSets which each contain the same number of members (i.e. have the samecardinality) are called equivalent sets. In this sense, the word equivalentrefers to the number of members; it does not apply to the type or nature ofthe objects in the sets.

The members of Set A can be matched one-to-one with the members ofSetB.

Set A and Set B have the same cardinal number.Set A is equivalent to Set B.

The classroom affords many opportunities of matching one object toanother. For example, giving one pencil to each child; 'Are there enoughchairs for the children at that table?' 'Give a piece of paper to each child onyour table.' 'Is there a girl partner for every boy?'

The amount of time given to mathematical activities with the youngestchildren simply cannot be measured if we agree that mathematics shouldnot be separated from the other areas of work. Mathematical vocabularyand experiences may arise at any time during the school day and it must beleft to the attitude and ability of the individual teacher as to how much orhow little she is able to extend the opportunities. Here are some furtherexamples of situations in which matching may be pinpointed:

Milktime A straw to each bottle, either one child doing this for a group ofchildren or each child matching his own; exchanging a picture symbol for a

15

Matching (N2)

bottle of milk; placing empty bottles in the crate-one bottle for eachspace; taking a handful of drinking straws and finding out whether thereare enough or too many for the bottles of milk.

Home corner Laying the table-a place setting for each child round thetable; 1 cup for each saucer; 1 cup for each hook; 1 lid for each saucepan; 1chair for each child.

Dressing up corner 1piece of clothing hung on each peg; 1 outfit for eachchild playing there.

Shop 1wrapping paper for each 'sweet' sold; 1 paper bag for each itemsold.

Cooking 1 apron to each cook; 1 spoonful of mixture to each paper case; 1paper case to each space in the patty-tin, 1 cake for each child; 1 cherry toeach cake; 1 spoonful of jam to each tart.

Physical education 1 blue ball for each boy; 1 yellow ball for each girl; 1hoop for each boy; 1 rope for each girl, etc.Eventually it will be necessary to contrive situations. For example, you

may provide a picture of a set of garages and a collection of cars made fromcard.

The pictures of the garages are cut where the doors meet so that a car canbe slotted into the opening.

The child is asked to putone car in each garage.Later he may be asked todraw a man for each car.

Further activities

slit

stout card

Eggs in egg cups cardboard egg shapes

16

slit

Flowers in vases

Matching (N2)

cardboard flowers

This is rather more difficult since more than one flower will fit into thevase.

Later a child may be asked to match up two sets by placing pieces ofstring or drawing chalk lines between each object and its partner to showwhether the sets are equivalent in number or not. If he is using concreteobjects he may wish to move them together, matching one from here withone from there. If the two sets are unequal, matching will help the child tosee which set has more and which has fewer members. This will give theteacher an opportunity to use the terms 'more than', 'less than', 'fewer','more'.A worthwhile activity is to ask a child to make a set to match one already

made. The child is given a string of different coloured square beads (squarebeads do not roll I),and asked to make a string of beads to match. Here thecolour is the important factor since the child matches each bead by itscolour.

Further examples, using concrete materials1 Matching beads to those drawn on a card by threading on to a lace (each

bead is of a different colour).

2a Matching bricks of different shapes to a set drawn on a card. 2b Matching shapes to those drawnon a circular shaped card.

17

Matching (N2)

3 Using a peg board and coloured pegs.

The teacher puts in a line of coloured pegs and asks the child to make aline underneath exactly the same. This is extended by giving the child acard with the set of pegs drawn on it. The child uses the card in order tomake a set of pegs on the peg board to match the set on the card.

R G e ~ II

#

R - r-edG -greenB -Dlu.e

4 The child has to make a string of beads to match one already threaded.Here all the beads are the same colour.

Some pictorial examples of matching-one-to-one correspondence Childrenenjoy using 'plug-in-cards'. These are made by pasting cut-out pictures ona stiff piece of card which may be covered with clear Contact for durability.A lace is threaded through a hole in each picture in one set and knotted atthe back of the card. Each lace can now be 'plugged in' to a hole made inthe corresponding picture in the other set.

Matcn one-to -one: -::-.--.-----.....-----,

18

Children go on to examples in which they match bydrawing a line between corresponding members, one ineach set. It is advisable to start with two setswhich are identical.

Matchthis set

.'

with this set.

Matching (N2)

Gradually, more members can be used in each set and the membersthemselves varied.

Match one-to-one.

Further examples are:

Match a set of: dogs to a set of bones.cats to a set of mice.monkeys to a set of bananas.rabbits to a set of carrots.knives to a set of forks.hats to a set of coats.bats to a set of balls.


~ Number 2.~lJ? sheetl.l

Match one-to-one.

oNow match them up.

2

sheet 1.2

19

Matching (N2)

N2: 2a Non-equivalent sets (two sets)This situation might already have arisen during earlier matching activitieswith real objects, but to be sure that children understand and are able touse terms such as 'more than', 'less than', or 'fewer' correctly, it isworthwhile introducing some pictorial exercises.Another version of the 'plug-in' card can be used with great effect to

show that two sets are not equivalent.

Matchthis set with this set.

Which set has more members?

Early examples should use similar looking objects in both sets in order toavoid confusion. Later, when matching ice-cream cornets to children,chairs to tables, cups to saucers, etc., follow-up questions such as, 'Arethere enough ice-creams?' 'Are there too many chairs?' should be asked.

Matchthis set with this set.

yesno

---..---....~.........---;--~~--..-,..-~

These ideas are further reinforced in the Bronto Book Bird has three eggs.

20

Is there an ice-cream for each girl?

Matching (N2)

N2: 2b Non-equivalent sets (three sets)Matching three sets in one-to-one correspondence in order to establishwhich set has least or most is merely an extension of the stage we havealready discussed. 'Plug-in' cards can be used again.

Match one-to-one.

Which nas most members?

Gradually by comparing three or more non-equivalent sets, children willlearn to order them, expressing their results in their own language, e.g.'There's more in this set', 'That set hasn't as many and this one is thelittlest.'

When possible the teacher should ask the children to describe the setseliciting such terms as 'most', 'more than' and 'least'. The sets should bearranged in both ascending and descending order according to the cardinalvalue of each set.

A suggested order of developmentis shown in the following illustrations:

Match one-to-one.

Match one-to-one.

Match one-to-one.Which set has the most member-s ?

Colour in that set.

21

Matching (N2)

Worksheets should not be introduced r========_~;:::::::::;:=::;:=;--,.' Number 2

until children have had plenty of ~ sheet 2.1practical experienceand opportunity for discussion.

~ Number2'. ~ sheet2.3

Match one-to-one

@Is there a spoon for each cup? Yes or no?

Is there a knife for each fork? Yes or no?

Is there Q bat for each ball? Yes or no?

3

Is there a knife and fork for each plate? Yes or no?

Is there a hat and coat for each boy? Yes or no?

6

22

sheet 2.6

Match one-to-one.fS\m~~Colour the set with more apples.

Match one-to-one.0:f>.~~0c&~~~)Colour the set with more members.

sheet 2.4

Match one-to-one

Are there fewer dogs than bones? Yes or no?

Are there more appLes than pears? Yes or no?

Are there more squares than triangles? Yes or no?

4

~ Number 2" . sheet 2.7OJ?

Match one-to-one.

Which set has the fewest members?Colour in the members of that set.

Which set has the most members?Colour in the members of that set.

7

sheet 2.2

Match one-to-one.

Colour the set with fewer bananas.

~ 1]4 ~ am ;1})C~~~ ~o BColour the set with fewer members.

2

sheet 2.5

Match one-to-one.

(J)~ c:J

CD ""'" u= 0

Matching (N2)

N2: 3 Pictorial representation and semi-tallyingThe mathematical value of pictorial representation is that it offers childrena visual interpretation of relationships which may not be obvious from awritten collection of data. It provides an opportunity for the teacher todiscuss with the children various relationships and to use the appropriatevocabulary. Many of the early sorting activities, for example, the numberof children going home to dinner, lend themselves to this sort ofrepresentation. With the teacher's help each child may represent himselfby placing a peg doll on a prepared chart. (Clothes pegs, which have beendressed, stand up if their 'feet' are stuck into a base of plasticine.)

westo.ya.tsc'000\ fo\0\ '(\'(\e'C.

If everyone is seen to place a doll on the chart, the children will begin toaccept that there must be the same number of dolls as there are children. Ifeach child has made and dressed his own doll, he can easily recognisehimself on the chart. Often other children pick out their friend's doll andsay 'Look you're there' or 'That's you.'

Later, bricks may be used to represent children whilst comparing adifferent relationship, e.g. the number of boys and girls in a group or class.The bricks may be placed in columns with suitable labels.

Other kinds of suitable materials Curtain rings placed on pieces of dowelrod; cotton reels either strung together or placed on dowelling; washers onknitting needles; poppet beads and plastic interlocking cubes may beuseful.

The important points at this stage are:1 That there is one object for every child involved in the work i.e. one-to-

one correspondence,2 That a comparison is made between two rows or columns only.

The graph activity serves as a link with 'semi-tallying' where objects orpeople are matched with cubes, poppet beads, counters, matchboxes,gummed squares, etc.

Put a counter for each cat"" in the bottom set.The arr-ows will help you.

23

Matching (N2)

Plastic interlocking cubes can be matched to the objects in a set, thenfixed together to make towers. The number of cubes in each tower willrepresent the number of objects in each set. The towers are then compared.'The tower for this set is taller than that one because there are more cubesin it.' This highlights the number of objects in the set and draws attentionto the particular property of a set which is called its cardinality or 'how-many-ness' .

Poppet beads may be strung together and used in a similar fashion,matching one bead to one object in each set. Here the length of the twostrings might be compared in order to establish which set has more objects.

Each time a child represents a person, object or event by a unit,comparison is easier and his attention is being focussed on the cardinalnumber of the sets. He is looking less at 'what the things in the set are like'and more at 'how many of them there are'. He is also working towards theidea of tallying using a more abstract representation.


24

Number 2sheet 3.1

sheet 3.2

TaUy. Draw a counter under each pig.

(~t;;;9)(__0 )

Tally. Draw a counter under each rabbit.

C~~e:v~)( )Tally. Draw a counter under each dog.

Cw ~ ~ ~ ~)( )

2

Matching CN2)

References and resourcesDewhirst, W. Bird has three eggs, Bronto Books Set D, Nuffield

Mathematics 5-11, Longman Group Ltd 1979

Frobisher, B. and Gloyn, S., Infants Learn Mathematics, Ward Lock 1969

Nuffield Mathematics Teaching Project, Beginnings 17, MathematicsBegins CD, Pictorial Representation QJ, Nuffield Guides,Chambers/Murray 1967 (See Introduction, page viii.)

Centicubes, Osmiroid

Metriblocs, Metricubes, Invicta Plastics

Multilink Linking Cubes, E.S.A.

Unifix Cubes, Philip & Tacey

Universal Miniature Rubber Stamps Sets 1 and 2, Philip & Tacey

25

Chapter 3

Counting andnumerals (H3)

26

For the teacherNumerals, the symbols or sounds which represent numbers, are all aroundus. Children meet them very early in their lives. They see them on buses,cars, house doors, gates, etc. Children use number words in their earlyspoken language but when a child says, 'It's my birthday today. I am four',it is doubtful whether he really understands to what thefour refers. He ismerely imitating a language structure he has heard adults use. He is usi~gthe word 'four' in a descriptive manner rather as he might say, 'I am tall',or 'I am sad.' The significance of numerals or number words will beappreciated gradually but, in the meantime, a child needs to be familiarwith their sound and appearance both in figure and word form. (Thenumber word 'three' and the figure 3 are both numerals since they bothstand for a number.)Finger plays, number rhymes and stories all help the child to become

familiar with the number names. The Bronto Book Counting with Brontoproduced by the Project, should also prove useful.

Summary of the stagesN3: 1 Counting

N3: 2 Matching a number symbol to a set

N3: 3 Introducing number words

N3: 4 The empty set

N3: 5 Conservation of number

Counting and numerals (N3)

VocabularyOne, two, three, four, five, six, seven, eight, nine, ten, zero, colour, count,draw, write, match, correct, missing numeral, number word.

Equiprnent and apparatusAbacus, peg numerals, jig saws, paper plates, small corn flake boxes usedfor sorting trays, bricks, plastic interlocking cubes, sorting trays.

3D reference set Square and round beads, clip on clothes pegs, table tennisballs, plastic toys, coloured plastic abacus tablets.

Conservation activities Matching and snap cards using differentarrangements of groups. Sew buttons on to cards or stick stars or otherpaper shapes on to cards.

Working with the childrenN3: 1 Verbal countingChildren may well be able to chant numerals in order but this does notnecessarily mean they can count. Learning to count comes gradually. Atfirst children may well count' 1, 2, 3, 4 ... ' getting quite out of step withthe objects concerned, or perhaps getting lost and counting some of themmore than once. It is not sufficiently recognized that there is a great gulfbetween the ability to recite the number words in sequence and the abilityto match the saying of the word to the moving of the object.

If children are given lots of activities, where objects 'counted' are movedphysically as the number word is spoken, then there will be less likelihoodof the child's finger overlapping two objects as he says one number name.

The large bead abacus found so often in a reception class is ideal for thissort of grouping. The child should be encouraged to move the bead acrossthe wire as he says the number name.

Sometimes it is helpful to provide sets of counters, bricks or beads ofdifferent colours, or a collection of quite different looking objects, makingit easier to remember which have been counted.When children begin to count, saying a number name aloud as they

touch or move an object, they are not usually aware of the cardinalnumbers. They mean the last number spoken to refer to the last objecttouched and not to the whole set. At this stage the counting is only verbal,with no operational value. To the child, 'five' is just a name for a certainobject. He does not see the group of objects as five beads, but as beads inisolation, the names of which are one, two, three, four and five. At thisstage children are not counting in an arithmetical sense by merely giving aname to each bead.

Children need lots of experience in counting different sets of objects, andquestions such as, 'How many objects have I altogether?' should be askedoften, so that the child begins to realize that the spoken numeral describesthe cardinality or 'how-many-ness' ofa set and not an individual member.

The teacher may wish to begin such an activity by placing two or threeobjects in her closed hand. She would then say to the child, 'I'm going to

27

Counting and numerals eN3)

Left side Right side

go that way go that way

0 2')-~ 3

~

16 t5)----s 1~ ~

qJ

28

open and close my hand quickly. See if you can tell me how many toys Ihave in my hand altogether.' When dealing with small numbers, childrenare usually able to give the correct number intuitively. As the number heldbecomes greater, the child will have to count. The idea of the activity isthat the child recognises that he is using the number name to describe thewhole set.

As the introduction to verbal counting is a practical activity -'there are noworksheets publishedfor this stage.

N3: 2 Matching a number symbol to a setAs children begin to count orally some of the activities will require thelabelling of objects with the symbol. Many of the symbols may berecognized but only as descriptive words. Their order in the numbersequence may not yet be known. Practice in writing the numerals can beincorporated in writing pattern work. To help the children remember thedirection in which each number symbol starts, their attention should bedrawn to the following facts:1 All numerals are written from the top.2 The numerals 0,4,6,8,9 are begun at the top and move towards the left.3 The numerals 2, 3, 5 and 7 are begun at the top and move towards the

right. (In the case of 5, the sequence is '.J' 5 .)Charts displaying this information will remind the children when theybegin to write.

The numeral 3 is preferred to 3 as it is less likely to be confused with 5.Most children find 4 easier to write than 4. However, children are likely tosee 3 and 4 so they must be made aware that these do not stand for differentnumbers but are just the way that some people write 3 and 4.Children should also be given the opportunity to write numerals in sand,

make them in plasticine or string and to 'feel' numerals cut out of glasspaper.

There are many types of match-and-count number recognition games onthe market but it is best to begin with apparatus that is self-corrective andlater move towards a situation which demands some knowledge ofnumerals and their value.

ActivitiesPeg numerals These are sets of numerals from 0 to 9 made out of woodabout 8 centimetres high. Each numeral acts as a peg board, with sufficient


holes in each to repesent its value. The child should count the pegs as heplaces them in each wooden numeral and later the numerals may be put inorder.

Jigsaw dominoes Home-made jigsaw dominoes. One half displays a groupof objects; the other half the matching numeral. Children should beencouraged to count the objects and then look for the correct numeral. Thetwo pieces will not fit together if the child counts wrongly or chooses thewrong numeral. Later, the child may be asked to put them in ordermatching them perhaps to a number line.

Sorting trays Paper plates or empty cornflake boxes may be painted,individually numbered and used as sorting trays. The teacher may putthem in order to begin with or ask the child to do it.

zero two

Counting board To help children to become familiar with numerals andtheir order, both Stern and Multilink have produced counting boardsconsisting of ten graded grooves into which the rods from 1 to 10 fit. Aboard displaying the numerals in order can be slotted into the back of theboard to be used by the child as a guide when he fits in a set of 10 numeralswritten on small blocks.

29


The apparatus is partly self-correcting but there is also a learningelement about it since, if the numeral board is left out, the child must findthe correct order in which to place the separate numeral blocks.

Unifix materials include a 1-10 Stair and 1-10 Value Boats which areboth self-correcting.

00o BgQae .o .

There are several versions of pattern boards produced which are alsoself-correcting.

'Read, match and count' 'Read, match and count' is a useful gameproduced by Gaits. It consists of two master cards divided into six equalboxes. In each is a numeral, a colour and a drawing of an object

e. g. 5 red ~ . There is also a set of cardsillustrating what is written in each box. These are not self-corrective, but

30


are easily matched if the child follows the clues.

Number strip A number strip, displaying sets of numbers from 1 to 10.The child counts out a small number of beads, say five, starting with 1 onthe number strip, he counts as he touches each numeral in order until hereaches the figure 5. He then pulls out that card and places it with his 5beads.

Number card A homemade number card displaying numerals will allowchildren to place the number of objects in correct order. This is not se1f-corrective and might be used as a checking-up exercise.

"2 6"II

I

-.5

Towers of Unifix bricks may be built and the correct 'numeral hat'placed on top. The hats come in the shape of roof tops e.g. ~ .

The towers too, can be put in sequence at the end of the activity. This isnot self corrective. Some teachers prefer to place the 'hats' besides eachtower as they find that occasionally a child counts in the 'hat' as a unit.

3

31


32

Circular sorting tray There is a strong plastic circular sorting tray on themarket which can be used in several ways. The middle compartment can beused to hold small toys or objects in which case numerals may be placed inthe surrounding compartments either at random or in order. The childcounts the correct number into each. Alternatively the teacher may placeone numeral, e.g. 3 in the middle and ask the child to put a different set ofthree objects in each compartment. Or the teacher may place a pack ofcards displaying the numbers from one to ten and ask the child to take onefrom the top, then place it in a compartment with the correct number ofobjects.

Recording counts by drawing

Draw atl the cowsitt the picture.

The child reads and copies writing into his book, and then carries out theinstruction ...

(This instruction may have to be read to the child.)


Draw 3 red f{owersand 1bl ue flower ..

dote .....

Match numeral to set.

sheet 2.4

Colour end count. Draw Q set of S.

C)5 5

Write in the missing numerals.

4

Colour in 4flowers.

DODDDOD

ODD 0Colour in 6 &1uares.

The card is covered in clear Contact and the child colours directly on tothe card using a china graph water base pen. This is easily rubbed off with adamp cloth.


06' Drow c set ot Z 007 7~~----------- ---------------

Match numeral to set.

2

Colour and count. Draw a set of 6.

Colour and count.

5

sheet 22 sheet 2.3

Colour and count.Match the sets that have the same number.

3

Draw a set of 6.

sheet 2.6

Colour and count.

Draw a set of 7.

6

33


Match numeral to set.Colour and count.

sheet 2.7

CoLour and count.

Draw a set of 8.

Draw a set of q. Draw a set of 8. Draw a set of q.

7 8

__________ --=sh:.::.::ee~t2:.:..:...:::.10 sheet 2.11

A busy picture. CoLour and write the numerals.

How many?

There are houses.

There are boys.

There are dogs.

There are

There are girls.

There are birds.

10

~@~

11

~ Number3~ sheet2.9

CoLour and count. Draw a set of 10.

Colour and count.

Write in the numerals.

9

sheet 2.12

Draw the correct number of beads.

10Write in the numerals.

12

N3: 3 Introducing the word form of numeralsThe words two, three, four etc., are also numerals and children will need alot of practice in matching a set of objects with the correct written numbername. Many reading schemes introduce the words one, two, three quiteearly; so it is likely that some children will have met these words already. Itis helpful, however, if the words are displayed alongside the figures. Whenbead threading, the cards threaded on to the lace could have both forms ofthe numeral written on them.

34

The picture value could be recordedon the reverse side.


tWO~3three 5five zone ~four

A three-dimensional reference set makes a useful display.

Each piece of strong card has a piece of thin elastic threaded over thefront to hold the appropriate number of objects such as: square arid roundbeads; clip-on clothes pegs (the coloured plastic ones are most attractive);table tennis balls (carefully pierced); coloured plastic abacus tablets with ahole; or sets of strong plastic toys (a hole can be made by using a heatedknitting needle).

Similar workcards to those described earlier may be used; the writtenword being substituted for the figure.

Trays or boxes with a number word in them may be filled with differentobjects by the children and changed daily.

Individual or class books may be made about a particular number e.g.,'My book about four'. A child may draw a picture ofa table and a chair forexample, as both have four legs. A favourite with the children is the 'Twobook'. Examples drawn seem endless-two shoes, plimsolls, gloves, eyes,socks, wellingtons, boots, arms, legs, ears, etc.Eventually children should be able to match numerals in words and

figures without any props.

Cards with the number word and the equivalent set of objects may bematched to the numeral. Similarly the number word and figure may bematched to the appropriate set. The idea is to use either the figure or set ofobjects alongside the number word.

Count and draw four beads.4- beads

count and draw four beads,4b~Qds

When finished.

35


Display tables about a certain number may be set up in the classroomand are a great favourite with the children. These should be changed oftenin order to introduce different values.


Number 3sheet 3.1

Draw and write the correct number word.

[JI one

1D0D011[IJI two IF.;[D:==m:t::;;;w-:::o==~M;;;;;j

1000 II[TIl three I F.;[TI~:t;:;flre::::::, ~e==;;;;;;:;;;;;'"I~eeell~I four I !=[fJ=;--f=--o-ur---====

sheet 3.3


66565{)(J

12]1seven 1[L] peven

(j{j{j{jOO{J{J

lJ1 eight I~-eight.

3

36

sheet 3.2


~~f;]J~f;]J1]]1 five I [[] five,

~OQ[JOB

@:JI six 1~r:5i;.~

2

sheet 3.4


OL)ODDt:)ODD

~Inine I~ nine ;"""

fJP'ofJJ)fJfJfJJ.)j:J

w,[!Q]I ten

4


has

ta-n ~@t-t ~@

N3: 4 The empty setThe idea of the empty set is difficult for young children to grasp but it isvery important. Each set has a number property and it is the number ofmembers in the empty set which is called 'zero', just as a set with fourmembers has a number property called 'four'. The children'sunderstanding of zero will be limited to : Question; 'How many sweetshave you left?' The child answers; 'None'.

Children need to encounter lots of situations dealing with the empty setin a physical sense, and discuss them with their teacher. For example, threechildren have a purse each, one child has two pennies in her purse, anotherhas three pennies, the third child's purse is empty.


Number 3sheet 4.1

Colour; count and read.~(!)OL~ '" j 0 means zero. C)

Count and label.

Join each set to its numeral.

cY({]~W2

o f 2 3 4- 5zero one two three four five

0234-5

3

sheet 4.3

Match the numerals to the sets. ~

O@~~(09\~~[i][!][i]w

sheet 4.2

2

Count and match to numeral.

sheet 4.4

t~~nine~

eight GI3seven 080six 00

five-r.:three~two one

@

4

zero

37

Counting and numerals CN3)

N3: 5 Conservation of numberA five or six-year-old. child may readily agree that there are the samenumber of counters in these two rows:

e e -- e -0 o G) CD @)If the second lot is spread out, he may say that there are more in the

second row than in the first:

eCD

eCD -CD

Even counting does not always help. A child may well say there are 5 ineach row but still assert that there are more in the second row! There is aconflict between what he sees (apparently more in the second row) andwhat he knows, i.e. that he has counted the same number in each row. Wesay a child can conserve number when he is quite sure that the number ofelements in a set remains constant no matter how they are arranged, i.e.whether close together or well spread out, or stacked in piles.

Children need lots of practice in matching one-to-one the members oftwo sets, then rearranging one set and matching them up again. Theinvariance of the number of elements in each set may be established byjoining each member of one set to a member of the other set with string,chalk marks or pencil lines. Further examples of this may be practised if weintroduce pictorial representations, e.g. the teacher draws a set of boys,four or five, and asks the child to draw a set of footballs, making sure thateach boy has one football. The set can then be matched by drawing linesbetween each boy and a football.Other examples are:a doll for a girl, a basket for each woman, a briefcase for each man, awhistle for each policeman.

Gradually the child will develop the idea of invariance of the number ofmembers in a set regardless of arrangement. But although he may be ableto conserve 5 or 6 members he may still be unable to recognize theinvariance of larger numbers. Accepting the invariance of number is agradual process and will run through much of the number work that thechild will do later. Much practical experience of pairing and matching one-to-one is nec~ssary before such ideas are firmly established.

Activities (conservation)1 There should be lots of matching and snap games using cards showing

different arrangements of certain groups. No numerals are used, thechild simply matches different arrangements of the same number.

38


2 Introduce numeral cards and match all arrangements to the correctnumeral.

Other cards may be made by sewing buttons on, or sticking on stars orother paper shapes, all in different arrangements. It is important that thesets of cards are colourful and well presented to make the gameappealing and interesting.

3 A child is asked to count out 7 coloured bricks. Then he arranges themspatially according to their colour, making a tower of each colour.

three fouryellow Dlue

one two onewh'lte yellow blue

two onegreen r-oo

He takes further sets of 7 bricks and continues to repeat the process,looking for as many different arrangements as he can find. The teachershould be at hand to discuss his work with him and ask questions suchas, 'Are there the same number of bricks in this set as there are in thatone ?' He may well think that the set of small towers has more bricks in itthan the set of tall towers. This can easily be disproved if the first set ismade into similar towers to the second set and the bricks matched one toone. However, this action alone will not establish invariance since oncethe bricks are again disarranged the child may lose sight of the bricks asa group and believe the set to have grown or diminished in number.Only by practising activities of the sort outlined above involvingcounting, re-arranging, matching and handling of 3D objects, willchildren develop the idea of conservation which is such an importantaspect of the concept of number.

4 Some commercially produced plastic cubes can be locked together in anon-linear fashion. This enables children to experiment with manydifferent formations, not necessarily in a straight line, which all containthe same number of cubes.

39



~ Number3~ sheet 5.1

Count. then put a cross X on the set thot does not match.

)$)J7~;!ffi~~~~

~S0:J~~JkColour the sets which match the numeral.

6

7

8

3

40

sheel5.3

sheelS.2

Colour the sets which match the numeral.

3 @@~~~~~

4

5

2

10

4


References and resourcesBurke, P., Counting with Bronto, Bronto Books Set A, Nuffield

Mathematics 5-11, Longman Group Ltd 1979

Frobisher, B. and Gloyn, S., Infants Learn Mathematics, Ward Lock 1969

Nuffield Mathematics Teaching Project, Beginnings W-, MathematicsBegins CD (Nuffield Guides), Chambers/Murray 1967 (See Introduction,page viii.)

Shuard, H. and Williams, E., Primary Mathematics Today (chapters 1-3),Longman Group Ltd 1970

Centicubes, Osmiroid

Jigged Number Plaques, E. J. ArnoldMetriblocs, Metricubes, Invicta Plastics

Multilink Counting Board, E.S.A.

Names and Numbers, Metric Aids

Peg Numbers, Metric Aids

Radial Plastic Sorting Trays, Philip & Tacey

Random Dominoes, E. J. ArnoldStern Counting Board and Pattern Boards, E.S.A.

Unifix 1-10 Boats and Pattern Boards, Unifix 1-10 Stair, Philip & Tacey

41

Chapter 4

Ordering (N4)

For the teacherAfter looking at the concept of ordering in a general sense, this chaptergoes on to deal with the ordering of non-equivalent sets. As a prelude tothis, children may have already begun to order length, capacities, shapes,events, etc., using the appropriate language such as 'longer than', 'holdsless than', 'is larger than', 'before' and 'after'. In the Number SectionN2: 2, sets will have been compared without specific reference to theircardinal value but on the basis that one set had more members thananother. This experience can now be carried over to enable the children tolook carefully at the number of elements in non-equivalent sets, label themaccordingly and put the sets into order.

Ordering at this stage has two aspects:

1 order by virtue of position in a line or a time sequence (first, second,last, etc.),

2 order by virtue of size or cardinal number.

In some cases, order occurs in a repeating pattern or cycle of events:

42

... Friday, Saturday, Sunday,Monday,Tuesday, .

Ordering (N4)

Where the pattern is not cyclic, however, it is important to indicate tothe children where to start and in which direction to go. For example, 'Putthese in order of size, starting with the smallest.' When using theworksheets, children should be encouraged to 'read' the pictures from leftto right as in reading words.A completed task would look like this:

Colour the third car.

~

Surrrmary of the stagesN4: 1 Ordinal numbers

N4 : 2 Putting non-equivalent sets in order

N4: 3 Signs 'is greater than', 'is less than'

N4: 4 Tallying and pictorial representation

VocabularyFirst, second, third, fourth ... etc., last, next to, in front of, behind, isgreater than, is more than, is less than, is fewer than, least, most, fewest,continue, column.

Equiprnent and apparatusThe following are useful:pictures of engine and six or seven carriages, mounted on darkbackground; wooden ladder, spinner, pipe-cleaner dolls; two sets of paperdoll's clothes, string and dowelling to make 'washing lines' ; hoops or loopsof string, beads, counters, poppet beads, cubes.

Working with the childrenN4: 1 Ordinal numbersMany children are helped at home by their mothers to acquire the ability tosay number words in sequence, e.g. counting fingers or toes at bath-time.Let us consider what happens when mother and child climb a short flightof stairs counting as they go. As the mother's foot touches each stair shesays a word, 'one ... two ... three, etc.' The child may be joining in withher. What are they really doing? They are giving a name to each stair. Atleast, that is the probable interpretation for the child. It will be a long timebefore the child will be able to tell you that, 'Stair three comes before stairfour and after stair two.' To emphasize that we are now interested in theorder, we could say, 'First ... second ... third, etc.' These are called

43

Ordering (N4)

'ordinal numbers'.Children find little difficulty in understanding first, second, third and

last since they often jostle with each other for these positions when liningup. Nevertheless words denoting position should be used as often aspossible by the teacher in real situations, e.g. today is the fourth of May;today is your fifth birthday; he's read his sixth book.

A worthwhile activity is to ask the children to help make a train with sixor sevencarriages (more can be added later). Each carriage must bedistinguishable from others either by colour or by name. The engine andcarriages are then mounted on an appropriate backing in such a way thatthey can easily be removed and placed in a different position. Whenmounted on the wall, the words and symbols for first, second, third, etc.,are placed in order beneath the carriages starting with the one nearest theengine.

The children discuss the order of the carriages and their position behindthe engine. Today the blue carriage may be first, the green carriage second,etc. Tomorrow after moving the carriages about the children may see thatthe red carriage is first, the blue second and perhaps the green carriage last.Children soon begin to realize that even though the carriages are movedabout, the name of the position in the order remains unchanged.

A simple activity which helps children to learn and recognize the wordsfirst, second, ... etc. and the symbols Ist, 2nd, ... etc. uses a card showinga line of animals:

Put a cou rrt.er on the I..:;~===;;J~ cow.

and a set of small cards which fit into the frame:

5th ] I sixth ~t=:==1 s~t-=-'Ji I second ] IA small card is selected, placed in the frame and the instruction followed

by another child.

44

Ordering CN4)

The procedure can be reversed. The teacher places a counter on a dogand the child selects the correct card(s).

Cards covered with clear adhesive plastic can be used repeatedly if thechild is given a felt tip pen for drawing. A damp cloth clears the card readyfor another child.

~eo~~~Draw the fifth animal.

Draw the sixth shape.

"first8(j~Draw the fourth face.

Many opportunities for using positional names will arise during P.E.lessons whilst playing games where the leader of a team or first in linequickly becomes second, third and so on, as other members of the team runfrom the end of the line and take their turn at being first.

Another simple game to help put over the idea of position may be playedwhen using six or seven hoops. These are set out in an agreed order andlabelled accordingly. The teacher arranges one child in each hoop and asksthe rest of the children which hoop a certain child is in, e.g. Question:'Which hoop is Derek in?' Answer: 'The second hoop/ If there are severalchildren in each hoop the question might be, 'How many children arestanding in the fourth hoop ?'

45

Ordering (N4)

A checking-up activity Two sets of doll's clothes are made out of paper.These can be coloured and decorated by the children. Two parallel clotheslines are set up between two short pieces of dowelling set in a base. Theteacher sets out one set of clothes on the top line. The child is asked to putthe second set of clothes on the bottom line, in exactly the same order. Thesecond part of the check is to ask the child to reverse the order of theclothes on the top line, as he places them on the bottom line. It is better ifeach set contains an odd number of items so that there is a middle one.

Cut clothes out on foldof paper. This waythey will hang on theline.

The test assesses the child's concept of order. Very often children whohave not achieved this level of mathematical understanding will deal withthe second stage of reversing the order of garments until about the middleof the clothes line. At this point, the child loses sight of his starting pointand matches the second half of his own line to the order set out on the topclothes line.


~Number4~ sheet 1.1

Colour the 3rd aeroplane.

~~_ II: I

II _ II

CoLour the 2nd house.

J'/I..,soldter ls on th e fint~ The dog IS on the

/1~~ The ball is on thed The boat is on then,. The Qer~Plane is on the11I.f"l;. The cat IS on the

~Od The teddy IS on the~ The car is on the

~ The elephant is on the

step.Colour the 4th boat. CoLour the Last boat.

step.

step.

~.... ~. ~~. ~.~. ~.~.. ~'''.' ..'..

~' .. .... ", ..- -. ,.- . .... I: ".

..., .. .step.step.

Colour the 5th teddy.step.

step.

step.Colour the 2nd ball. Colour the 8th ball.

step.

2

46

sheet 1.2

Ordering (N4)

N4: 2 Putting non-equivalent sets in order according to theircardinal valueFrom the moment a child begins to recite the number jingle he is orderingnumerals although the activity has little meaning for him. Only when hebegins to match numerals to sets and put them in order is he beginning tolook at the special way in which our number system is built. Throughample practice of matching words or symbols to the members of a set, thechild will gradually come to realize the order in which natural numbersgrow:

There is a constant difference of one between each number and the nextin sequence,each number value has a position on a number line,whilst being one greater than the number it follows, each number is alsoone less than the number it precedes.

Initially, the children need to compare three sets only. Each set shouldcontain members which are easily identified and bear no resemblence to themembers of the other two sets.For example:

A set of cat5A set of hen'S

The sets are made by sticking pictures or printing rubber stamps onseparate cards or by placing sorting toys on paper plates.Through discussion and matching or counting, the children decide:

Which set has the most members?Which set has the least (or fewest) members?Are there more dogs than hens?Are there fewer frogs than hens? etc.

Eventually, a child can be asked to move the sets so that they are in order,sometimes ascending, sometimes descending.Very gradually, changes or refinements are introduced into the activity

so as to reduce the difference between sets to just one member and toincrease the number of sets being ordered. The aim should be to progresstowards ordering sets containing 0 to 5 members, each displaying a

47

Ordering (N4)

difference of one from the previous set. The ability to order up to 5 can bechecked by using a card:

Fill in the missing set.

and a selectionof card pieces:

The child selects the correct set to place in the loop. (It is important tomake the set loops all the same shape and size so that the child does notselect by 'fitting' the piece of card.)When the child is confident about ordering small numbers, greater

numbers up to 10may be introduced using a counting board (see Chapter3) or threaded beads:

48

Ordering CN4)

Peg boards are useful for demonstrating the way in which numbersgrow. The numerals should be stuck in order along the bottom edge of thepeg board and the child asked to insert the correct number of pegs in eachcolumn. The child's attention should be drawn to the number of pegs in aparticular column, comparing its number value with the columns bothbefore and after, for example 'Three comes after two and in front of four. ,

Putting a set of rods in order bymaking a staircase within a traymay help the child to see thenumber values increasing ordecreasing by one each time.

The use of a number ladder may help towards the understanding ofnumber relationships. The ladder should be solidly constructed so that thechild may prop it against a wall. Two dolls can be made from pipe cleaners,these will wrap easily round the rungs.

Make a spinner or dice on which different scores are drawn. Colour 'ups'red; 'downs' blue; 'No move' white. To start, place both dolls on the samerung about halfway up. After three moves, children take account of thesituation, which might be :

Girl doll (Jane) is on number 5 rung.Boy doll (John) is on number 3 rung.

John needs to take two steps to reach Jane, so 5 is greater than 3.

49

Ordering (N4)

~ Number4~ sheet 2.1

There ore more birds than

There are more than birds.

There are fewer cups than

There are fewer than cu ps.

Count and colour the correct number.

888888888888BBBBBBBBB8888888888888888888~~~99~~999[i]~~~~@][?J~~~

4

50

sheel2.4


sheet 2.2

.Fill in the missing set. ~

2

sheet 2.5

Draw the missing sets. Write in the numerals.

C])C])0D@OODO DO. DO DO

~00006 8

5

~Number4~ sheet2.3

Colour 3 cups.

Colour 6 trees.

oDO

ODDDODD

DDDDDDDOOOO

DDDDDDDDDDDDDDD

DODD DODODDDDDDDDDDD

Count, colour and Label.

3

10

6

N4: 3 Using symbols for 'Greater than' and 'Less than'When a child begins to notice differences he should be encouraged to usethe appropriate language 'longer than' or 'shorter than' when comparinglengths, 'holds more than' or 'holds less than' when comparing capacity,etc. When comparing the cardinality of sets, that is comparing the numberof members of one set with the number of members of another set, theexpressions 'greater than' (or 'more than') and 'less than' (or 'fewer than')are used.

As a child's skill in language grows, it is likely that his ability to record inlonghand will develop. To begin with, recording should be brief andenjoyable and not detract from the pleasure of the experience. At this stage,the written word will only supplement what has already been expressedorally by the child. The child will still be discussing his findings with his

Ordering CN4)

teacher, since only through discussion with individual children will theteacher be able to assess development, and any writing will be a simplerecord of the child's achievements.

Initially, children will record in simple graph form, using drawings ofthemselves, coloured squares, beads or stamps, etc. As children begin toconsider number relationships in words,

8 is more than 7 9 is less than 10

it would seem appropriate to introduce the shorthand symbols:

8 >7 9 < 10

However, these symbols must not be introduced too early and should onlybe used to compare numbers.Before introducing them, the teacher should check that the child is able

to count, that he is completely familiar with the sequence of numbers to tenand has some sense of both the ordinal and cardinal aspects of number.

The symbol for 'is greater 'than' is >5 > 3 (five is greater than three).

The symbol for 'is less than' is and < is to use semi-tallying:

---

51

Ordering (N4)

This idea may also be introduced by using cubes and cards similar to these:

>

As a check-up, cards similar to this may be used:

52

>

Ordering (N4)


Number 4sheet 3.1

Write in the missing numerals.

I4.. ::::>

IS greater than is greater than

is greater than is greater than

than

sheel3.3

5> 3 means 5 is greater than 3.4- means "is greater than".< means "is less than",

Write in the missing Signs.

5 is greater than 3.or 5 3

6 is greater than 2.or6 ::> 2

4- is less than 6.~ c::::: 6

Write and read. Write and read.

Put in > or < f

Nuffield Maths Teacher's Handbook 1

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Transcript of Nuffield Maths Teacher's Handbook 1