041532498 x Trouble With Maths

The Trouble with Maths

This book offers important insights into the often-confusing world of numeracy. By ­looking­at­learning­difficulties­in­maths­from­several­perspectives,­including­the­language­ of­mathematics,­thinking­styles­and­the­demands­of­individual­topics,­Steve­Chinn­delivers­ a­comprehensive­text­which­will­become­an­essential­classroom­companion­to­anyone­who­ uses it. Whilst­considering­every­aspect­concerning­maths­and­learning,­this­book­provides­a­ perfect­balance­of­advice,­guidance­and­practical­activities,­enabling­the­reader­to: ♦­­ develop­flexible­thinking­skills; ♦­­ use­alternative­strategies­for­pupils­to­access­basic­facts; ♦­­ implement­effective­preventative­measures­before­disaffection­sets­in; ♦­­ recognise­maths­anxiety­and­tackle­self-esteem­problems; ♦­­ make­accurate­ongoing­assessments­of­pupils’­difficulties; ♦­­ design­informal­diagnostic­procedures. With­useful­features­such­as­checklists­for­evaluation­of­books,­software­and­test­materials,­ this book highlights essential skills that will allow teachers to diagnose and address maths difficulties­ and­ improve­ standards.­ It­ draws­ on­ tried­ and­ tested­ methods­ based­ on­ the­ author’s­years­of­classroom­experience­to­provide­an­authoritative,­yet­highly­accessible­ one-stop­ classroom­ resource­ for­ all­ teachers,­ classroom­ assistants,­ Special­ Educational­ Needs­Co-ordinators,­student­teachers,­and­learning­support­staff.

Steve Chinn­is­Principal­and­Founder­of­Mark­College,­Somerset,­a­specialist­school­for­ dyslexics­awarded­‘Beacon­school­status’­by­the­Department­for­Education­and­Skills.

The Trouble with Maths

A­practical­guide­to­helping­learners­with­numeracy­difficulties

Steve­Chinn

2­Park­Square,­Milton­Park,­Abingdon,­Oxon,­OX14­4RN Simultaneously­published­in­the­USA­and­Canada

by­RoutledgeFalmer 270­Madison­Ave,­New­York­NY­10016

RoutledgeFalmer is an imprint of the Taylor & Francis Group

This­edition­published­in­the­Taylor­&­Francis­e-Library,­2009. To purchase your own copy of this or any of

Taylor­&­Francis­or­Routledge’s­collection­of­thousands­of­eBooks please go to www.eBookstore.tandf.co.uk. ©­2004­Steve­Chinn All­rights­reserved.­No­part­of­this­book­may­be­reprinted­or reproduced­in­any­form­or­by­any­electronic,­mechanical, or­other­means,­now­known­or­hereafter­invented,­including photocopying­and­recording,­or­in­any­information­storage or­retrieval­system,­without­permission­in­writing­from the publishers.

British Library Cataloguing in Publication Data

A­catalogue­record­for­this­book­is­available­from­the British Library

Library of Congress Cataloging in Publication Data

A­catalog­record­has­been­requested ISBN­0-203-35809-0­Master­e-book­ISBN

ISBN­0-203-47769-3­(Adobe­ebook­Reader­Format) ISBN­0-415-32498-X­(Print­Edition)

Contents

List of illustrations vi

Foreword ix

1­ Introduction:­learning­difficulties­in­mathematics­ 1

2 Factors that affect learning 20

3­ What­the­curriculum­asks­pupils­to­do­and­where­difficulties­

may occur 33

4 Thinking styles in mathematics 52

5­ Developmental­perspectives­ 69

6­ The language of maths 86

7 Anxiety­and­attributions­ 97

8­ The inconsistencies of maths 105

9­ Assessment and diagnosis 111

10­ The­nasties…long­division­and­fractions­ 130

Appendix 1 Further Reading 142

Appendix 2 Checklists and resources 143

Appendix 3 Jog Your Memory cards for multiplication facts 146 Appendix 4 Setting an inclusive maths department policy 150

Appendix 5 Criterion referenced tests 155

Notes 165

Illustrations

Figures

2.1 Place­value­cards

30

2.2 A blank task analysis form

31

2.3 An­example­of­a­task­analysis

32

3.1 Using­10p­coins­as­images­of­number­bonds­for­100

39

3.2 Base­ten­blocks­used­to­show­23×12

42

3.3 Area­model­for­31×17

43

3.4 28×39­compared­to­30×40­as­areas

44

3.5 Division­by­subtraction­of­‘chunks’

45

4.1 Subtraction­the­inchworm­way

53

4.2 What­is­the­area­of­the­shaded­figure?

55

4.3 Trial and adjust the inchworm way

59

5.1 Area models for multiplication

72

5.2 The­blank­table­square

74

5.3 The­almost­complete­table­square­with­the­‘easy’­number­

_{facts shaded}

₇₅

5.4 6×10­compared­to­6×9

76

5.5 Grouping repeated additions into partial products

77

5.6 Multiplication­and­division­using­the­‘easy’­numbers

79

5.7 Images­of­number­bonds­for­10

81

5.8 Number­bonds­for­100­with­base­ten­blocks

82

6.1 Place­value­arrow­cards

87

7.1 Expectations

101

9.1 Seating­for­the­assessment

116

9.2 Dot­recognition­and­counting

118

9.3 Place­value­cards

121

9.4 Screening­test­for­basic­numeracy­skills

122

9.5 A­non-diagnostic­worksheet­for­a­Level­2­pupil

129

10.1 ‘Traditional’­long­division

130

10.2 The criss-cross-times method for fractions

134

10.3 The­hidden­division­sign

136

10.4

136

10.5 Fractions and clocks

139

10.6 Renaming­fractions­to­make­them­have­the­same­name

140

Tables

4.1 Thinking styles of the inchworm and the grasshopper

52

Foreword

For­many­years,­the­field­of­children’s­learning­difficulties­has­been­dominated­by­research­on­ reading­and­dyslexia.­There­is­now­a­large­body­of­knowledge­concerning­the­nature­and­origins­ of­dyslexia­and­how­best­to­teach­reading­to­children­with­such­difficulties.­However,­learning­ to­read­is­not­the­only­skill­that­poses­difficulty­for­children­with­dyslexia-related­disorders—the­ majority­also­have­difficulty­with­at­least­some­aspects­of­number­work.­‘The­Trouble­with­ Maths’­addresses­this­issue­and­shows­how­all­who­teach­and­learn­maths­can­enjoy­it. Scientific­research­on­number­skills­has­increased­during­the­past­ten­years­and­the­evidence­ base­is­growing­at­a­rapid­rate.­It­is­now­known­that,­from­shortly­after­birth,­infants­can­detect­ changes­in­the­number­of­objects­in­visual­displays­of­objects.­This­innate­sense­of­numerosity­is­ the­foundation­of­later­number­concepts­that­develop­rapidly­during­the­pre-school­years.­But­to­ become­numerate­demands­more;­children­need­to­learn­conventional­systems­and­to­use­their­ mathematical­thinking­meaningfully­and­flexibly­in­logical­situations.­Neuroscientists­mapping­ brain-behaviour­relationships­have­shown­that­mathematical­expertise­draws­on­a­wide­range­of­ neural­circuits­in­the­right­and­left­hemispheres­that­must­work­in­concert­to­solve­mathematical­ problems.­It­is­therefore­not­surprising,­as­Steve­Chinn­points­out,­that­children’s­problems­with­ numeracy­are­diverse­and­the­term­‘dyscalculia’­may­be­too­broad­to­capture­the­wide­range­of­ problems­that­are­observed­in­children­acquiring­mathematical­skills. At­the­broadest­level,­numeracy­problems­may­affect­either­mathematical­thinking­or­com-putational­skills.­But­during­development,­as­this­book­shows,­the­situation­is­never­quite­so­ simple;­mathematical­learning­is­a­cumulative­process­so­that­a­poorly­developed­concept­of­ number­can­affect­the­acquisition­of­number­facts­and­by­the­same­token,­poor­arithmetic­abil- ity­can­compromise­the­growth­of­mathematical­knowledge.­Furthermore,­school­mathemat-ics­is­embedded­in­language,­requires­sequencing­skills,­draws­on­working­memory­systems,­ depends­on­spatial­concepts­and,­more­than­any­other­school­subject,­engenders­anxiety.­Add­to­ this­the­fact­that­the­assessment­of­mathematical­learning­usually­requires­the­child­to­read­and­ comprehend­written­questions­and­to­lay­out­their­work­clearly,­and­the­whole­area­become­a­ minefield­for­the­child­with­specific­learning­difficulties. Steve­Chinn­has­done­a­great­job­of­systemising­what­can­be­done­to­help­the­child­who­ has­problems­with­maths.­His­message­is­clear—take­account­of­the­strengths­of­the­child­and­ the­style­of­their­thinking,­marry­this­with­the­demands­of­the­numeracy­curriculum­and­adjust­ lessons­to­help.­The­book­shows­how­careful­assessment­can­lead­to­the­development­of­effec-tive­strategies­to­help­with­basic­and­more­advanced­concepts­of­maths.­There­is­discussion­of­ what­data­mathematics­tests­can­provide­(and­by­inference­what­they­cannot)­and­a­checklist­for­ choosing­assessment­instruments.­There­are­heaps­of­ideas­for­successful­teaching­and­advice­ on­both­how­to­set­up­an­individual­education­plan­and­how­to­develop­an­inclusive­mathemat-ics curriculum. Written­from­the­perspective­of­a­highly­skilled­practitioner­and­an­expert­teacher­educator­ who­enjoys­and­understands­numbers,­this­book­ensures­that­all­who­read­it­will­understand­ how­to­teach­maths­in­such­a­way­that­every­child­can­achieve­their­potential. Maggie­Snowling­ University­of­York­ February 2004

Chapter­1­

Introduction

Learning­difficulties­in­mathematics

This­book­is­to­help­teachers,­classroom­assistants­and­learning­support­assistants­who­have­ to­deal­with­pupils­who­are­underachieving­in­mathematics.­It­takes­several­perspectives­ on­the­situation,­from­preventive­measures­to­diagnosis­and­identification­of­difficulties,­to­ thinking­styles,­to­ideas­for­intervention.­It­works­like­a­repair­manual­in­some­respects­and­ like­a­care­awareness­manual­(looking­after­your­students)­in­other­respects. It­is­a­book­which­can­be­accessed­in­different­ways.­It­can­provide­an­overview­of­ where­and­how­problems­may­arise.­It­offers­insights­into­areas­of­potential­difficulty.­It­ can­focus­on­a­particular­problem­and­suggest­approaches­which­can­help­the­pupil­to­learn,­ but­it­would­be­an­impossible­task­to­attempt­to­provide­an­answer­for­every­problem­for­ every­child. It­can­be­used­to: ♦­­ identify­a­problem ♦­­ understand­possible­reasons­for­a­problem ♦­­ pre-empt­problems ♦­­ develop­flexible­thinking­skills ♦­­ circumvent­problems­in­basic­numeracy ♦­­ address­the­difficulties­pupils­have­with­word­problems ♦­­ teach­alternative­strategies­for­accessing­basic­facts ♦­­ recognise­maths­anxiety,­attributional­style­and­self-esteem­problems ♦­­ design­informal­diagnostic­procedures ♦­­ extract­diagnostic­information­from­pupils’­work ♦­­ stimulate­ideas­for­teaching­maths­to­pupils­who­are­facing­difficulties­with­the­ subject. Sometimes­you­may­find­information­repeated­in­different­chapters­of­the­book.­This­is­ deliberate­as­some­observations­fit­into­more­than­one­area.­The­new­area­should­give­a­ different­perspective­to­that­information.

What do learners need to be good at mathematics?

Although­ this­ book­ is­ about­ what­ to­ do­ when­ learners­ are­ underachieving­ in­ maths,­ it­ should­be­valuable­to­consider­what­learners­need­to­be­good­at­maths.­I­have­two­sources­ for­this­information.­One­is­from­the­USSR­and­the­other­from­the­USA.

Krutetskii1 _{presented a broad outline of the structure of mathematical abilities during}

school­age.­He­specifies:

♦­­ The­ability­for­logical­thought­in­the­sphere­of­quantitative­and­spatial­relationships,­ number­and­letter­symbols;­the­ability­to­think­in­mathematical­symbols.

♦­­ The­ability­for­rapid­and­broad­generalisation­of­mathematical­objects,­relations­and­ operations.

♦­­ Flexibility­of­mental­processes­in­mathematical­activity.

♦­­ Striving­for­clarity,­simplicity,­economy­and­rationality­of­solutions.

♦­­ The­ ability­ for­ rapid­ and­ free­ reconstruction­ of­ the­ direction­ of­ a­ mental­ process,­ ­switching­from­a­direct­to­a­reverse­train­of­thought.

♦­­ Mathematical­memory­(generalised­memory­for­mathematical­relationships),­and­for­ methods­of­problem­solving­and­principles­of­approach.

♦­­ These­components­are­closely­interrelated,­influencing­one­another­and­forming­in­their­ aggregate a single integral syndrome of mathematical giftedness.

Although­Krutetskii­makes­these­observations­concerning­giftedness­in­mathematics,­they­ are­equally­appropriate­for­competence.­The­reader­can­see­where­learning­difficulties­may­ create problems. The­other­source­is­the­National­Council­of­Teachers­of­Mathematics­in­the­USA,­who­ list­and­explain­twelve­essential­components­of­essential­maths: 1­­ Problem solving­The­process­of­applying­previously­acquired­knowledge­to­new­and­ unfamiliar­situations.­Students­should­see­alternate­solutions­to­problems:­they­should­ experience­problems­with­more­than­a­single­solution.

2 Communicating mathematical ideas (receiving and presenting)­Students­should­learn­ the language and notation of maths.

3­ Mathematical reasoning­ Students­ should­ learn­ to­ make­ independent­ investigations­ of­ mathematical­ ideas.­They­ should­ be­ able­to­ identify­and­extend­ patterns­ and­ use­ ­experiences­and­observations­to­make­conjectures.

4 Applying maths to everyday situations­Students­should­be­encouraged­to­take­everyday­ situations,­translate­them­into­mathematical­representations­(graphs,­tables,­diagrams­ or­mathematical­expressions),­process­the­maths­and­interpret­the­results­in­light­of­the­ initial situation.

5­­Alertness to the reasonableness of results­In­solving­problems,­students­should­question­ the reasonableness of a solution or conjecture in relation to the original problem. They must­develop­number­sense.

6­­ Estimation­Students­should­be­able­to­carry­out­rapid­approximate­calculations­through­ the­use­of­mental­arithmetic­and­a­variety­of­computational­estimation­techniques­and­ decide when a particular result is precise enough for the purpose in hand.

7 Appropriate computational skills­ Students­ should­ gain­ facility­ in­ using­ addition,­ ­subtraction,­multiplication­and­division­with­whole­numbers­and­decimals.­Today­long­ complicated­computations­should­be­done­with­a­calculator­or­a­computer.­Knowledge­ of single digit number facts is essential.

8­­ Algebraic thinking­Students­should­learn­to­use­variables­(letters)­to­represent­mathematical­ quantities­and­expressions.­They­should­understand­and­use­correctly­positive­and­nega-tive­numbers,­order­of­operations,­formulas,­equations­and­inequalities.

9­­Measurement­Students­should­learn­the­fundamental­concepts­of­measurement­through­ concrete­experiences.

10­ Geometry­ Students­ should­ understand­the­ geometric­ concepts­necessary­to­function­ effectively­in­the­three­dimensional­world.

11­­Statistics­Students­should­plan­and­carry­out­the­collection­and­organisation­of­data­to­ answer­questions­in­their­everyday­lives.­Students­should­recognise­the­basic­uses­and­ misuses of statistical representation and inference.

12­­ Probability­ Students­ should­ understand­ the­ elementary­ notions­ of­ probability­ to­ ­determine­the­likelihood­of­future­events.­They­should­learn­how­probability­applies­to­ the decision making process.

And,­picking­up­on­Krutetskii’s­first­point­concerning­the­use­of­symbols­in­maths,­the­ ­British­psychologist,­Skemp2_­wrote: Among­the­functions­of­symbols,­we­can­distinguish: (1)­Communication (2)­Recording­knowledge (3)­The­communication­of­new­concepts (4)­Making­multiple­classification­straightforward (5)­Explanations (6)­Making­possible­reflective­activity (7)­Helping­to­show­structure (8)­Making­routine­manipulations­automatic (9)­Recovering­information­and­understanding (10)­Creative­mental­activity. He­concludes­that:­‘It­is­largely­by­the­use­of­symbols­that­we­achieve­voluntary­control­ over­our­thoughts.’ So,­we­have­some­characteristics­for­being­good­at­mathematics.­Could­one­assume­ that­deficits­in­all­or­some­of­these­skills­create­difficulties­in­mathematics?­I­know­from­ experience­that­there­are­many­reasons­why­someone­may­underachieve­in­mathematics­ and­that­the­picture­is­a­complex­one­with­no­single­root­cause.­Recently­the­term­dyscal-culia has become more prominent and so a look at what dyscaland­that­the­picture­is­a­complex­one­with­no­single­root­cause.­Recently­the­term­dyscal-culia might be may help our ­understanding­of­other­reasons­for­difficulties­with­mathematics.

Dyscalculia, definitions and descriptions

Dyscalculia,­a­problem­with­learning­mathematics,­is­attracting­attention­in­official­circles.­ The­DfES­(Department­for­Education­and­Skills)­has­published­a­booklet­(see­below)­on­ guidance­for­supporting­pupils­with­dyscalculia­(and­dyslexia)­in­the­National­Numeracy­ Strategy­(NNS).­There­is­now­a­screening­test­for­dyscalculia3_{­published­by­NFER-Nelson.­} Despite­this­attention­the­literature­on­dyscalculia­is­sparse­and­the­definitions­are­a­little­ bland­at­present.­I­have­extracted­a­few­from­various­sources. Developmental­dyscalculia­is­defined­by­Bakwin­and­Bakwin­(1960)­as­a­‘difficulty­ with­counting’­and­by­Cohn­(1968)­as­a­‘failure­to­recognise­numbers­or­manipulate­them­ in­an­advanced­culture’.­Gerstmann­(1957)­describes­dyscalculia­(Gerstmann’s­syndrome)­ as­ ‘an­ isolated­ disability­ to­ perform­ simple­ or­ complex­ arithmetical­ operations­ and­ an­ impairment­of­orientation­in­the­sequence­of­numbers­and­their­fractions.’

Kosc­(1974)­describes­developmental­dyscalculia­as­a­structural­disorder­of­mathematical­ abilities which has its origin in a genetic or congenital disorder of those parts of the brain

that are the direct anatomico-physiological substrate of the maturation of the mathematical abilities­ adequate­ to­ age,­ without­ a­ simultaneous­ disorder­ of­ general­ mental­ functions.­ This­definition­clearly­puts­dyscalculia­in­the­inherited­and­specific­learning­difficulties­ category. In­1996­Magne­gave­a­slightly­more­cautious­explanation­of­a­difficulty­in­mathematics­as­ the­low­achievement­of­a­person­on­a­certain­occasion­which­manifests­itself­as­performance­ below­standard­of­the­age-group­of­this­person­or­below­his­own­abilities­as­a­consequence­ of­inadequate­cognitive,­affective,­volitional,­motor­or­sensory­etc.­development.­The­cause­ for­inadequate­development­may­be­of­various­kinds.­This­description­acknowledges­that­ there­will­be­more­than­one­cause­for­difficulties­in­mathematics.

Mahesh­ Sharma­ (1986)­ lists­ the­ many­ words­ that­ have­ been­ suggested­ for­ maths­ ­difficulties,­explaining­terms­such­as­acalculia,­dyscalculia,­anarithmetica­and­noting­that­ there­is­no­definite­agreement­on­their­use­universally­in­the­literature,­that­they­have­not­ been­used­consistently,­and­although­there­are­significant­differences­between­dyscalculia­ and­acalculia,­some­authors­have­used­the­terms­interchangeably.­He­concludes­that­the­ descriptions­of­these­terms­are­quite­diverse­to­say­the­least. Sharma­suggests­that­dyscalculia­refers­to­a­disorder­in­the­ability­to­do­or­to­learn­math- ematics,­that­is,­difficulty­in­number­conceptualisation,­understanding­number­relation-ships­and­difficulty­in­learning­algorithms­and­applying­them.­It­is­an­irregular­impairment­ of­ability.­Thus­Sharma­suggests­that­dyscalculia­is­a­specific­learning­difficulty. Acalculia­is­used­to­label­a­more­serious­condition,­the­loss­of­the­fundamental­pro-cesses­of­quantity­and­magnitude­estimation­and­a­complete­loss­of­the­ability­to­count.­ This­is­an­acquired­condition. And,­finally,­something­that­sounds­like­a­missing­member­of­the­Russian­royal­family,­ arithmastenia,­defined­as­a­uniform­deficiency­in­the­level­of­mathematical­abilities. I­wrote­the­following­for­the­Dyslexia­Institute’s­journal,­Dyslexia Review,­volume­14,­ number­3,­2003.­It­is­reprinted­with­their­permission.

Does dyscalculia add up?

Initial ramblings

Is dyscalculia ‘dyslexia with moths’?

With­the­publication­of­Brian­Butterworth’s­The Dyscalculia Screener and the inclusion of­dyscalculia­as­a­specific­learning­difficulty­in­the­DfES­consultation­document­for­the­ 2004­SEN­census,­dyscalculia­is­a­hot­topic.­This­article­sums­(!)­up­my­current­thinking­ about­dyscalculia.­Unfortunately­my­current­thinking­is­fluid.­I­am­trying­to­make­sense­of­ all­those­factors­which­influence­the­maths­learning­outcomes­of­children­and­adults.­So,­I­ hope this paper may attract some responses and stimulate more research.

Since­absolute­knowledge­on­dyscalculia­is­in­short­supply­I­am­going­to­construct­this­ paper­around­the­questions­which­I­consider­we­have­to­investigate­to­reach­an­understanding­ of­dyscalculia.­In­doing­this­there­seem­to­be­some­very­interesting­comparisons­between­ dyscalculia­and­dyslexia.

­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ There­are­some­things­I­know­as­a­start.­I­know­that­dyscalculia­will­not­be­a­simple­ ­construct­(I­think­that­means­a­psychological­concept).­I­know­that­there­will­be­many­ reasons­why­a­person­may­be­bad­at­maths.­I­know­there­will­not­be­any­instant­or­simple­ ‘cures’­because­I­know­that­there­is­unlikely­to­be­a­single­reason­behind­the­problem­of­ the­many,­many­people­who­fail­to­master­maths­and­I­know­that­not­all­of­these­will­be­ dyscalculic. I­heard­David­Geary­speak­at­the­last­IDA­conference.­This­American­guru­compared­ our­ knowledge­ of­ dyslexia­ to­ being­ close­ to­ adulthood­ and­ our­ knowledge­ of­ maths/­ dyscalculia­to­being­in­its­early­infancy.­This­is­reflected­in­the­number­of­research­studies­ done­on­language­difficulties­compared­to­those­done­on­maths­difficulties.­As­for­studies­ on­dyscalculia,­they­are­few­indeed.­I­think­there­are­so­many­parallels­at­so­many­levels­ between­dyslexia­and­dyscalculia­and­all­that­surrounds­these­specific­learning­difficulties,­ for­example­prevalence,­definition,­teaching­methods,­etiology­and­so­forth.

We­ are­ some­ twenty­ years­ behind­ language/­dyslexia­ studies­ in­ our­ knowledge­ and­ understanding­of­dyscalculia.­This­is­not­to­say­that­I­think­it­will­take­us­twenty­years­ to­catch­up­in­all­areas,­but­that­it­takes­a­good­length­of­time­for­the­concept­to­become­ accepted­in­everyday­educational­settings­and­thus­for­understandings­to­build­from­work­ from­the­‘shop­floor’. So,­let’s­go­back­twenty­years­to­a­much­quoted,­pioneering­paper­by­Joffe­(1980).­One­ of­Joffe’s­statistics­has­been­applied­over-enthusiastically­and­without­careful­consideration­ of­how­it­was­obtained.­This­is­the­‘61­per­cent­of­dyslexics­are­retarded­in­arithmetic’­and­ thus­39­per­cent­are­not.­Now­it­is­not­quite­as­simple­as­that.­The­sample­for­this­statistic­ was­ quite­ small,­ some­ 50­ dyslexic­ learners.­The­ maths­ test­ on­ which­ the­ statistic­ was­ largely­based­was­the­British­Abilities­Scales­Basic­Arithmetic­Test­which­is­just­that,­a­ test­of­arithmetic­skills.­Although­the­test­was­untimed,­Joffe­noted­that­the­high­attainment­ group­would­have­done­less­well­if­speed­was­a­consideration.­The­extrapolations­from­ this­paper­would­have­to­be­cautious.­Other­writers­seem­to­have­overlooked­Joffe’s­own­ cautious­and­detailed­observations,­for­example,­‘Computation­was­a­slow­and­laborious­ process­for­a­large­proportion­of­the­dyslexic­sample.’­You­will­see­at­the­very­end­of­this­ paper­I­have­mentioned­a­Mark­College­pupil­who­was­identified­as­dyscalculic­by­the­ Butterworth­screener­(where­two­out­of­the­five­exercises­focus­on­speed­and­accuracy­in­ computation)­but­who­is­predicted­to­achieve­a­Grade­A­in­GCSE­maths. I­think­there­are­two­reasons­why­Joffe’s­paper­is­so­frequently­quoted.­One­is­that­it­is­a­ good­paper­and­the­other­is­that­there­are­so­few­others­from­which­to­quote.­NFER-Nelson­ describe­Brian­Butterworth­as­‘the­leading­expert­on­dyscalculia’.­Sadly,­at­the­moment­it’s­ a­one­horse­race.­We­need­more­researchers­to­follow­Butterworth’s­initiative.

Definitions and labels

As­a­(lapsed)­physicist­I­have­a­scientist’s­concept­of­what­makes­a­definition.­In­physics­ one­can­control­the­variables­and­do­pretty­reliable­experiments.­People­are­difficult­to­ control­(especially­as­teenagers).­In­this­respect­I­view­some­definitions­as­descriptions.

The­ definition­ of­ a­ learning­ difficulty­ can­ be­ very­ influential­ and­ can­ have­ many­ ­consequences.­For­example,­it­can­influence­the­allocation­of­resources­to­an­individual­or­ to­a­school­or­to­the­education­budget­of­an­Education­Authority.­For­an­individual,­knowing­

that­your­difficulties­have­a­label­may­be­a­relief­and­a­benefit,­but­it­may­also­cause­a­ ­reaction­not­at­all­dissimilar­to­that­of­grieving­for­a­loss.­So­there­needs­to­be­a­sense­of­ responsibility­and­awareness­of­all­these­implications­in­those­who­create­definitions. There­seems­to­have­been­a­change­in­the­culture­of­the­definition­of­dyslexia,­from­ the­all­encompassing­definitions­of­the­late­80s­to­the­focused,­minimalist­definition­of­the­ British­Psychological­Society­in­the­late­90s.­This­could­well­be­significant.­Professor­Tim­ Miles­talks­of­‘lumpers’­and­‘splitters’.­So,­could­there­be­‘specific­learning­difficulties’­ which­may­encompass­all­or­some­of­dyslexia,­dyspraxia­and­dyscalculia­or­can­the­three­ dys’s­have­independent­existences?

Does being dyscalculic exclude you from being dyslexic or dyspraxic? Does being dyslexic exclude you from being dyscalculic? Then, turning to the lumpers, does being dyslexic mean you are also dyscalculic and dyspraxic?

It­may­help­to­answer­some,­if­not­all­of­these­questions­if­you­think­of­real­people,­real­ individuals­and­what­the­answer­would­be­for­Jeff­or­Jane.­My­feeling­is­that­the­answer­to­ the­first­two­questions­is­‘No’­and­to­the­third­question,­‘Not­necessarily.’ Of­course­the­answers­depend­on­the­definitions­currently­assigned­to­the­difficulties.­ I’ll­come­back­to­summarise­my­thoughts­on­definitions­towards­the­end­of­this­paper,­but­ here­is­a­small­sample­of­definitions­for­now.

The­ first­ is­ from­ a­ DfES­ booklet­ (2001)­ on­ supporting­ pupils­ with­ dyslexia­ and­ ­dyscalculia­in­the­NNS.

Dyscalculia­ is­ a­ condition­ that­ affects­ the­ ability­ to­ acquire­ mathematical­ skills.­ Dyscalculic­ learners­ may­ have­ difficulty­ understanding­ simple­ number­ concepts,­ lack­an­intuitive­grasp­of­numbers,­and­have­problems­learning­number­facts­and­ procedures.­Even­if­they­produce­a­correct­answer­or­use­a­correct­method,­they­may­ do­so­mechanically­and­without­confidence. Very­little­is­known­about­the­prevalence­of­dyscalculia,­its­causes,­or­treatment.­ Purely­dyscalculic­learners­who­have­difficulties­only­with­numbers­will­have­cogni-tive­and­language­abilities­in­the­normal­range,­and­may­excel­in­non-mathematical­ subjects.­It­is­more­likely­that­difficulties­with­numeracy­accompany­the­language­ difficulties­of­dyslexia. The­second­dates­from­1970­and­is­attributed­to­Kosc­(1974:47). Developmental­dyscalculia­is­a­structural­disorder­of­mathematical­abilities­which­ has its origin in a genetic or congenital disorder of those parts of the brain that are the direct anatomico-physiological substrate of the maturation of the mathematical ­abilities­ adequate­ to­ age,­ without­ a­ simultaneous­ disorder­ of­ general­ mental­ functions.

Sharma­(1990)­discusses­three­terms­for­difficulties­in­mathematics,­saying­that, Terms­such­as­acalculia,­dyscalculia,­anarithmetica…there­is­no­definite­agreement­ on­their­use­universally­in­the­literature…they­have­not­been­used­consistently… although­there­are­significant­differences­between­dyscalculia­and­acalculia,­some­ authors­have­used­the­terms­interchangeably…the­descriptions­of­these­terms­are­ quite­diverse­to­say­the­least. He­explains­dyscalculia­and­acalculia­as:

Dyscalculia­ refers­to­a­disorder­in­the­ability­to­do­ or­to­learn­mathematics,­that­ is,­difficulty­in­number­conceptualisation,­understanding­number­relationships­and­ difficulty­ in­ learning­ algorithms­ and­ applying­ them.­ (An­ irregular­ impairment­ of­ ability). ­­­­Acalculia­is­the­loss­of­fundamental­processes­of­quantity­and­magnitude­estima-tion.­(A­complete­loss­of­the­ability­to­count.) The­final­example­is­from­a­DfES­consultation­document­called­‘Classification­of­SEN.’­ The­descriptions­are­to­be­used­in­the­pupil­level­annual­schools­census­from­2004­(DfES­ 2002).

Specific­ learning­ difficulty­ (SpLD)­ covers­ a­ range­ of­ related­ conditions­ which­ occur­across­a­continuum­of­severity.­Pupils­may­have­difficulties­in­reading,­writ-ing,­spelling­or­manipulating­numbers­which­are­not­typical­of­their­general­level­of­ ­performance.­Pupils­may­have­difficulty­with­short-term­memory,­with­organisational­ skills,­with­hand-eye­coordination­and­with­orientation­and­directional­awareness.­ Dyslexia,­dyscalculia­and­dyspraxia­fall­under­this­umbrella.

Pupils­ with­ dyscalculia­ have­ difficulty­ with­ numbers­ and­ remembering­ ­mathematical­facts­as­well­as­performing­mathematical­operations.­Pupils­may­have­ difficulties­ with­ abstract­ concepts­ of­ time­ and­ direction,­ recalling­ schedules­ and­ sequences­of­events­as­well­as­difficulties­with­mathematical­concepts,­rules,­formu-las­and­basic­addition,­subtraction,­multiplication­and­division­of­facts.

What distinguishes dyscalculia from just problems with maths? What do we mean by ‘prob-lems with maths’? How big is the problem?

We­don’t­know.­It­will­depend­on­the­definition.­It­may­also­depend­on­the­persever-ance­of­the­difficulty.­Goodness­knows­how­many­people­have­a­‘difficulty’­with­maths.­ Like­all­skills,­if­you­cease­to­practise­you­lose­the­skill­and­few­adults­practise­maths,­ ­especially­topics­such­as­fractions­or­algebra,­after­leaving­school.­So­the­extent­of­the­ problem­could­well­increase­in­adults.­For­example,­a­study­done­in­1995­on­behalf­of­the­ Basic­Skills­Agency­of­1714­adults­aged­35­years­found­that­just­under­one­quarter­had­ very­low­numeracy­skills­at­a­level­which­would­make­it­difficult­to­complete­everyday­ tasks successfully.

So I am sure that just having a difficulty with maths should not automatically earn you the label ‘dyscalculic’.

Dyscalculia­introduces­another­word­into­the­vocabulary­of­special­needs.­Some­see­ these words as labels and thus as descriptors of a person. That would not be helpful.

OK, I’m dyscalculic. So what?

I­like­the­questions,­‘What­if?’­and­the­follow­up­‘So­what?’­‘What­if­I­am­dyscalculic,­ so­ what?’­ I­ have­ to­ ask­ does­ being­ dyscalculic­ condemn­ the­ learner­ to­ being­ forever­ ­unsuccessful­at­maths.­That­then­raises­further­questions:

‘What does it mean to be successful at maths?’ and What skills and strengths does a learner need to be successful at maths?’ and ‘Is it important to be successful at maths?’

At­my­school,­Mark­College,­a­DfES­approved­independent­school­for­boys­who­have­ been­diagnosed­as­dyslexic,­the­results­for­GCSE­maths­are­significantly­above­national­ average.­Usually­at­least­75­per­cent­of­grades­are­at­C­and­above­compared­to­the­national­ average­of­around­50­per­cent.­Obviously­I­believe­that­if­the­teaching­is­appropriate­then­ a­learning­difficulty­does­not­necessarily­mean­lack­of­achievement.­But,­does­a­C­grade­ or­above­in­GCSE­maths­define­success?­That’s­a­question­for­another­article,­so,­for­the­ purpose­of­this­article­let’s­assume­it­is­one­criterion­and­let’s­assume­this­is­one­piece­of­ evidence­that­appropriate­teaching­can­make­a­difference. As­for­maths,­well­there­is­the­maths­you­need­for­everyday­life.­This­rarely­includes­ algebra,­fractions­(other­than­ and ),­co-ordinates­or­indeed­much­of­what­is­taught­ in­secondary­schools.­It­does­include­a­lot­of­money,­measurement,­some­time­and­the­ ­occasional­ percentage.­Take,­ as­ an­ example­ of­ a­ real­ life­ maths­ exercise,­ paying­ for­ a­ ­family­meal­in­a­restaurant.­It­needs­estimation­skills,­possibly­accurate­addition­skills,­ subtraction­skills­if­using­cash,­and­percentage­skills­for­the­tip.

The­ Russian­ psychologist­ Krutetskii­ (1976)­ listed­ the­ components­ of­ mathematical­ ­ability­which­could­act­as­a­description­of­what­a­learner­needs­to­be­‘good­at­maths’­and­ thus­also­act­as­a­guide­as­to­what­may­be­the­deficits­which­handicap­the­learner­failing­to­ be­good­at­maths­(also­listed­on­page­21ff­in­more­detail). 1­­ An­ability­to­formalise­maths­material­(to­abstract­oneself­from­concrete­numerical­ relationships). 2­­ An­ability­to­generalise­and­abstract­oneself­from­the­irrelevant. 3­­ An­ability­to­operate­with­numerals­and­other­symbols. 4­­ An­ability­for­sequential­segmented­logical­reasoning. 5­­ An­ability­to­shorten­the­reasoning­process. 6­­ An­ability­to­reverse­a­mental­process. 7­­ Flexibility­of­thought.

8­­ A­mathematical­memory. 9­­ An­ability­for­spatial­concepts.

What is maths?

Could a person be good at some bits of maths and a failure at other bits? Do you have to fail at ALL bits to be dyscalculic?

In­ terms­ of­ subject­ content,­ early­ maths­ is­ mostly­ numbers.­ Later­ it­ becomes­ more­ ­varied­ with­ new­ topics­ introduced­ such­ as­ measure,­ algebra­ and­ spatial­ topics.­ Up­ to­ GCSE,­despite­the­different­headings,­the­major­component­remains­as­number.­So­the­ demands­of­maths­can­appear­quite­broad,­and­this­can­be­very­useful,­but­number­can­be­ a­disproportionate­part­of­early­learning­experiences.

So poor number skills could be a key factor in dyscalculia. This might suggest that we have to consider the match between the demands of the task and the skills of the learner.

In­terms­of­approach,­maths­can­be­a­written­subject­or­a­mental­exercise.­It­can­be­ formulaic­or­it­can­be­intuitive.­It­can­be­learnt­and­communicated­in­either­way,­or­com-bination of ways by the learner and it can be taught and communicated in either way or combination of ways by the teacher.

Maths­can­be­concrete,­but­fairly­quickly­moves­to­the­abstract­and­symbolic.­It­has­ many rules and a surprising number of inconsistencies.

In­terms­of­judgement,­feedback­and­appraisal,­maths­is­quite­unique­as­a­school­sub-ject.­Work­is­usually­a­blunt­‘right’­or­‘wrong’­and­it­has­to­be­done­quickly.

Even on this brief overview it is obvious that the demands of maths are varied. The importance placed on speed of working could also be another key issue for learners.

Attitude and the affective domain

I­don’t­have­the­reference,­but­there­was­a­study­done­in­Scandinavia­which­summed­up­the­ influences­of­language­and­maths­skills­on­life.­Excuse­me­if­I­state­the­influences­somewhat­ starkly.­It­is­important­to­remember­that­people­do­not­have­to­follow­the­conclusions­of­ statistical­analysis.­Being­good­at­English­does­not­predict­success­in­life.­Being­bad­at­ English­predicts­failure.­Being­bad­at­maths­does­not­predict­failure.­Being­good­at­maths­ predicts success.

Of course we all know that being bad at maths holds no social stigma in Western culture. Indeed­it­may­well­attract­much­mutual­empathy.­So­the­consequences­of­dyscalculia­are­ going­to­have­a­better­social­acceptance­than­the­consequences­of­dyslexia.­(For­example,­ I­recently­read­a­letter­to­the­Times­about­a­restaurant­menu,­complaining­that­since­it­had­ spelling­mistakes­the­writer­would­not­be­eating­there.­That­makes­sense). Schools,­of­course­rarely­reflect­life.­In­school­there­may­well­be­significant­consequences­ of­being­bad­at­maths,­for­example­the­allocation­of­the­learner­to­a­teaching­group­which­

may­limit­the­levels­of­work­in­several­other­subjects.­Also­in­school,­unlike­life,­it’s­hard­ to­avoid­the­stuff­you­don’t­like­or­the­work­you­feel­you­can’t­do.

Two key factors which aid learning are ability and attitude. The latter can go a long way­towards­compensating­for­the­former,­but­then­the­two­factors­are­pretty­closely­inter-linked,­for­example­when­success­encourages­good­attitude. Some­learners­just­feel­that­they­can’t­do­maths.­This­may­well­be­a­consequence­of­early­ unsuccessful­learning­experiences­or­feedback­which­is­seen­as­negative.­The­judgemental­ nature­of­maths,­together­with­the­culture­of­having­to­do­maths­quickly­can­lead­children­ to­avoid­the­risk­of­being­wrong­again­and­again­and­thus­to­disassociate­themselves­from­ the­learning­experience­(Chinn,­1995).­Maths­creates­anxiety­and,­sadly­it­usually­seems­to­ be­an­anxiety­that­does­not­facilitate­learning.­Ashcraft­et al.­(1998)­has­shown­that­anxiety­ in­maths­can­impact­on­working­memory­and­thus­depress­performance­even­more. Some learners develop an attributional style for maths which makes their attitude personal as in ‘I’m too stupid to do maths’, pervasive, ‘I can’t do any maths’ and permanent, ‘I’ll never be able to do maths.’ An individual with a combination of those three beliefs could well present as a dyscalculic.

Memory, short, long and not always working

I­often­pose­the­question­in­my­lectures­‘What­does­the­learner­bring?’­(to­maths).­I­have­ already­mentioned­some­factors­such­as­anxiety.­But­what­about­memory?­I­know­that­ ­Krutetskii­(1976)­lists­mathematical­memory­as­a­requirement­to­be­good­at­maths.­I­am­ sure­that­short­term­and­working­memory­are­vital­for­mental­arithmetic,­particularly­for­ those­sequential,­formula­based­maths­thinkers.

But can a learner compensate for difficulties in some of these requirements and thus ‘suc-ceed’ in maths? Now­let’s­go­back­to­school,­in­England,­where­we­have­the­excellent­National­Numer-acy­Strategy.­This­truly­is,­in­my­opinion­an­excellent­programme,­but­however­excellent­ the­programme,­it­is­virtually­impossible­to­meet­the­needs­of­every­learner.­An­essential­ part­of­the­NNS­in­the­early­years­of­education­is­mental­arithmetic.­Now­that’s­an­activity­ that­needs­memories,­long,­short­and­working.­So­a­learner­with­a­poor­short­term­memory­ could­fail­maths­when­it­is­mental­maths,­even­though­he­may­have­the­potential­to­become­ an­effective­mathematician.­If­failure­is­internalised­as­a­negative­attributional­style­by­the­ learner­then­that­potential­may­never­be­realised. Is­Krutetskii’s­mathematical­memory­a­parallel­with­Gardner’s­multiple­intelligences?­ Perhaps­ there­ are­ multiple­ memories.­ That­ would­ explain­ some­ of­ the­ discrepancies­ I­ have­seen­in­children’s­memory­performances.­Like­any­subject,­there­is­a­body­of­factual­ information for maths and if a learner can remember and recall this information then he will­be­greatly­advantaged­and­if­he­can’t…

So good memories may be required for doing maths in general. Short term and working memories may be essential for mental maths and mathematical long term memory will be essential for the number facts and formulae you need when doing mental arithmetic.

Counting on and on

The­ first­ number­ test­ on­ the­ Butterworth­ Dyscalculia Screener is for subitizing. This means­an­ability­to­look­at­a­random­cluster­of­dots­and­know­how­many­are­there,­without­ ­counting.­Most­adults­do­this­at­6­years­old­plus­or­minus­one­year. A­person­who­has­to­rely­entirely­on­counting­for­addition­and­subtraction­is­severely­ handicapped­in­terms­of­speed­and­accuracy.­Such­a­person­is­even­more­handicapped­ when­trying­to­use­counting­for­multiplication­and­division.­Often­their­page­is­covered­in­ endless­tally­marks­and­often­they­are­just­lined­up,­not­grouped­as­ ­that­is,­in­fives.­ Maths­is­done­in­counting­steps­of­one.­If­you­show­them­patterns­of­dots­or­groups,­they­ prefer the lines and lines.

It’s­not­just­the­ability­to­‘see’­and­use­five.­It’s­the­ability­to­see­nine­as­one­less­than­ ten,­to­see­6+5­as­5+5+1,­to­count­on­in­twos,­tens­and­fives,­especially­if­the­pattern­is­not­ the­basic­one­of­10,­20,­30…but­13,­23,­33,­43…

It’s the ability to go beyond counting in ones by seeing the patterns and relationships in numbers­(Chinn­and­Ashcroft­1992).

Garden variety or what?

How­do­we­distinguish­between­a­‘garden­variety’­poor­reader­and­a­dyslexic?­(Stanovich,­ 1991)­ How­ do­ we­ distinguish­ between­ a­ ‘garden­ variety’­ poor­ mathematician­ and­ a­ ­dyscalculic?

I think the answer has a lot to do with perseveration of the difficulty in the face of skilled and varied and appropriate intervention.

Can­you­be­a­good­reader­and­still­be­a­dyslexic?­Can­you­be­good­at­some­areas­of­ maths­and­still­be­dyscalculic?­My­guess­is­that­the­answer­to­both­questions­is­‘Yes’,­but­ for­maths­it­is­partly­because­maths­is­made­up­of­topics,­some­of­which­make­quite­dif- ferent­demands­(and­for­both­questions,­good­appropriate­teaching­can­make­such­a­dif-ference).

Once again I drift back to problems with numbers as being at the core of dyscalculia. And it is numbers that will prevail in real life, when algebra is just a distant memory. And I guess that the main problem is in accessing these facts accurately and quickly, usually straight from memory, rather than via strategies.

Could­there­be­a­parallel­between­phonics­and­number­facts?­For­example­knowing­how­to­ use­phonics­to­spell­a­word­could­be­compared­to­using­addition­facts­to­add,­say,­572+319.

But­then­not­all­factors­are­intellectual.­A­difficulty­may­be­affected­by­a­bureaucratic­ decision.­Some­bureaucrats­specify­a­level­of­achievement­that­defines­whether­or­not­a­ child’s­learning­difficulties­may­be­addressed­or­even­assessed,­influenced­in­this­decision,­at­ least­in­part­by­economic­considerations.­But,­even­then,­is­a­child’s­dyslexia­or­dyscalculia­ defined­solely­by­achievement­scores?­Is­there­room­to­consider­the­individual­and­what­he­ brings­to­the­situation?­Sometimes­these­decisions­are­being­de-humanised.­So,­I­foresee­ a­child­not­receiving­provision­for­dyscalculia­unless­his­maths­age­is­five­or­more­years­

Teaching

I­claimed­that­being­a­physicist­influenced­the­way­I­think.­I­am­also­a­teacher­and­have­ been­for­almost­40­years­and­those­years­have­certainly­influenced­the­way­I­think,­too.­ The­teacher­part­of­my­thinking­says,­among­other­things,­‘So­he’s­dyscalculic,­what­do­ you­expect­me­to­do­next?’ Well,­my­guess­is­that­using­the­range­of­methods­and­strategies­we­have­developed­ at­Mark­College­for­teaching­our­dyslexic­pupils­will­also­be­effective­with­dyscalculic­ pupils.­Indeed­we­have­probably­taught­many­pupils­who­have­the­comorbid­problems­of­ dyslexia­and­dyscalculia.­What­we­address­as­teachers­is­the­way­the­pupil­presents,­not­a­ pupil­defined­by­some­stereotypical­attributes. My­colleague,­Julie­Kay­when­faced­with­a­learner­who­is­struggling­with­learning­maths­ asks­herself­the­questions,­‘Where­do­I­begin?­How­far­back­in­maths­do­I­go­to­start­the­ intervention?’­This­may­be­a­difference,­should­we­need­one,­between­the­dyscalculic­and­ the­dyslexic­who­is­also­bad­at­maths.­It­may­be­that­the­starting­point­for­the­intervention­is­ further­back­in­the­curriculum­for­the­dyscalculic­than­for­the­dyslexic.­Yet­another­topic­to­ research.­It­may­also­be­that­the­subsequent­rates­of­progress­are­different.­Another­topic. And­for­a­final­thought­in­this­section,­I­ask,­‘What­is­the­influence­of­the­style­of­curric-ulum?’­I­know,­for­example,­from­a­European­study­in­which­I­was­involved­(Chinn­et al., 2001),­that­the­design­of­the­maths­curriculum­certainly­affects­thinking­style­in­maths.

So what?

There are many reasons why a child or an adult may fail to learn maths skills and knowledge. For­example,­a­child­who­finds­symbols­confusing­may­have­been­successful­with­mental­ arithmetic,­but­may­find­written­arithmetic­very­challenging.­There­may­be­other­examples­ of an onset of failure at different times which will most likely depend on the match between the­demands­of­the­curriculum­and­the­skills­and­deficits­of­the­learner,­for­example,­a­dys-lexic­will­probably­find­word­problems­especially­difficult­and­a­child­who­is­not­dyslexic,­ but­is­learning­at­the­concrete­level­may­find­the­abstract­nature­of­algebra­difficult.­A­child­ who­is­an­holistic­learner­may­start­to­fail­in­maths­if­his­new­teacher­uses­a­sequential­and­ formula­based­inchworm­teaching­style.­A­learner­may­have­a­poor­mathematical­memory­ and­the­demands­on­memory­may­suddenly­exceed­his­capacity.

A difficulty will depend on the interaction between the demands of the task and the skills and attitudes of the learner. For example, if one of the demands of mental arithmetic is behind­the­norm,­which­could­mitigate­against­early­intervention­for­six­year­old­pupils.

that it be done quickly, then any learner who retrieves and processes facts slowly will have learning difficulties. Learning difficulties are obviously dependent on the learning task.

And­none­of­the­underlying­contributing­factors­I­have­discussed­are­truly­independent.­ Anxiety,­for­example­is­a­consequence­of­many­influences.­I­am­hypothesising­that­the­ factors­I­have­mentioned­are­the­key­ones.­There­may­well­be­others­and­the­pattern­and­ ­interactions­ will­ vary­ from­ individual­ to­ individual,­ but­ these­ are­ my­ choices­ for­ the­ ­difficulties­at­the­core­of­dyscalculia.

Of­the­definitions­I­have­quoted,­I­much­prefer­the­NNS­version.­The­DfES­consultation­ paper­version­seems­to­describe­just­difficulties­in­maths­which­might­occur­in­anyone.­ I­ have­ added­ some­ extra­ notes­ into­ the­ definition­ which­ may­ then­ be­ better­ seen­ as­ a­ ­description­(and­thus­not­a­label).

Dyscalculia is a perseverant condition that affects the ability to acquire mathematical skills despite appropriate instruction. Dyscalculic learners may have difficulty understanding simple number concepts­(such­as­place­value­and­use­of­the­four­operations,­+­−­×­and­÷),­ lack an intuitive grasp of numbers­(including­the­value­of­numbers­and­understanding­and­ using­the­inter-relationship­of­numbers),­and have problems learning, retrieving and using quickly number facts­(for­example­multiplication­tables)­and procedures­(for­example­long­ division).­Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence­(and­have­no­way­of­knowing­or­checking­that­the­ answer­is­correct).

The­NNS­version­focuses­on­number,­which­makes­sense­to­me.­It­mentions­memory­ and­it­includes­those­who­present­as­competent­in­some­areas,­but­whose­performance­ has no underlying understanding of number. An addendum could list some of the key ­contributors,­such­as:

A learner’s difficulties with maths may be exacerbated by anxiety, poor short term memory, inability to use and understand symbols, and inflexible learning style.

Now­the­definition/­description­is­in­this­form,­it­may­be­possible­to­set­up­a­diagnostic­ procedure.­That­sets­another­research­task­and­we­desperately­need­more­research!

Finally,­ have­ I­ met­ any­ learners­ whom­ I­ think­ would­ be­ described­ accurately­ as­ perseverently and exclusively­dyscalculic?­I­have,­but­they­were­few.­I­mention­two,­one­ is­a­female,­gifted­in­language­(and­languages)­who­had­absolutely­no­idea­what­ (presented­as­symbols)­would­be.­I­asked­her­would­the­answer­be­bigger­or­smaller­than­ 50­and­she­replied­‘Yes’.­The­other­is­a­male,­average­at­language­skills­but­who­could­ not­‘see’­that­I­held­out­three­fingers.­He­had­to­count­them,­even­as­a­sixteen­year­old.­He­ achieved­a­Grade­G­in­GCSE­maths. There­are­many­others­out­there­who­may­present­as­dyscalculic­as­young­learners.­It’s­ what­happens­next­that­confirms­or­challenges­that­description.

To add one piece of evidence to support the ‘What happens next’ hypothesis, I used Butterworth’s­ The­ Dyscalculia­ Screener­ on one of our Year 10 students.­ The­ Screener­ diagnosed him as dyscalculic. We are predicting he will achieve an A grade in GCSE

‘perseverent’.

If­ you­ want­ to­ follow­ up­ references­ for­ this­ section,­ they­ are­ listed­ at­ the­ end­ of­ this­ chapter. For­me,­the­main­issue­here­is­that­not­every­child­or­adult­who­is­failing­in­mathematics­ is­dyscalculic.­Even­for­those­who­do­gain­this­label,­it­does­not­predict­an­outcome,­but­it­ does­suggest­to­me­that­whatever­teaching­experiences­this­dyscalculic­pupil­has­had,­they­ have­not­been­appropriate.­I­know,­from­ten­years­of­data­on­pupils­at­Mark­College,­that­ it­is­possible­for­most­pupils­to­change­a­history­of­gains­in­maths­age­of­less­than­twelve­ months­per­year­to­gains­of­over­twelve­months­per­year,­thus­moving­to­‘catch­up’.­Some­ of­these­pupils­might­have­been­diagnosed­as­dyscalculic,­some­might­not.­In­many­senses­ that­was­less­relevant­than­their­history­of­underachievement­in­mathematics. And­would­intervention­for­a­dyscalculic­be­different?­The­answer­is­that­any­difficulty­ has­to­be­viewed­individually,­but­that­the­core­principles­of­teaching­and­learning­will­ probably­be­drawn­from­the­same­compendium­of­ideas­used­for­dyslexics­(see­Chinn­and­ Ashcroft­1998).

Adjusting lessons to help pupils who are having difficulties in learning

maths

Adjustments­to­lessons­should­be­based­on­four­principles:

♦­­ Empathetic classroom management­which­implies­an­active­awareness­and­consequent­ adjustment­to­the­learning­strengths­and­difficulties­of­pupils.

♦­­ Responsive flexibility­allows­the­teacher­to­have­a­repertoire­of­resources­and­strategies­ which­respond­to­the­individual­(and­often­changing)­needs­of­the­pupil.

♦­­ Developmental methods­are­methods­that­address­the­remedial­need­whilst­developing­ mathematical skills and concepts.

♦­­ Effective communication which infers an awareness of thinking and learning style and an­awareness­of­limitations­such­as­language­skills,­poor­short­term­memory­or­slower­ speeds of working.

The­application­of­these­principles­should­affect­all­levels­of­work,­from­the­construc-tion of the syllabus and lesson plans to the setting and marking of homework.

Integrating dyscalculic pupils and other learners who have

difficulties with mathematics into the real world of the

classroom

The syllabus and programme of work

1­­ A­structure­or­programme­which­builds­in­regular­returns­to­topics­helps­learners­with­ poorer­long­term­memories.­Frequent­revisions­and­overviews,­especially­after­a­short­ time­lapse­for­reflection­help­to­reinforce­learning­(largely­implicit­in­the­NNS). maths next year. Are these two statements incompatible? I think the answer lies in the word

2­­ Programmes­ that­ rely­ heavily­ on­ self­ tuition­ can­ allow­ pupils­ to­ develop­ incorrect­ ­procedures­ and­ concepts.­ (I­ remember­ the­ Kent­ Mathematics­ Project­ where­ pupils­ worked­largely­with­work­cards­and­at­their­own­pace,­but­with­little­or­no­tuition.) In the classroom

­

Short­ term­ memory­ deficits­ can­ affect­ mental­ arithmetic­ skills­ (which­ may­ show­ a­ marked­difference­to­written­arithmetic­skills). ­ ­Short­ term­memory­deficits­can­affect­many­other­areas­of­learning­such­as­the­­number­ of­items­of­instruction­a­pupil­can­process­at­one­time.­These­deficits­may­be­auditory­or­ visual­or­both,­so­presentation­should­always­address­both­modes. ­ Look­out­for­short­term­memory­overload­(when­the­pupil­will­just­be­overwhelmed­and­ recall­nothing­at­all). ­ ­ Reading­ deficits­do­not­affect­all­areas­of­mathematics­to­the­same­degree­and­are­a­ good­example­of­a­deficit­that­gives­rise­to­a­seemingly­inexplicable­change­in­level­of­ performance­(that­is,­when­word­problems­are­introduced). ­

Some ­pupils­are­slower­to­produce­work,­due­to­such­factors­as­writing­speed, poor­ organisational­skills,­finger­counting­instead­of­instant­recall­of­facts.­Speed­of­working­ is­often­an­issue­in­mathematics­and­can­be­the­cause­of­greatly­increased­anxiety­and­ greatly­ decreased­ levels­ of­ performance.­ Consider­ allowing­ extra­ time­ for­ tests­ and­ examinations.­Consider­careful­selection­of­quantities­of­work­for­these­pupils.

­Anxious­learners­are­often­poor­risk­takers­and­will­not­try­work­they­perceive­to­be­ ­difficult,­thereby­avoiding­failure­(they­have­usually­had­enough­experience­of­failure),­ but­ they­ are­ then­ not­ accessing­ new­ learning­ experiences.­ Research­ in­ the­ 1920s­ showed­ that­ a­ pupil’s­ first­ experience­of­ applying­ new­ knowledge­ is­ the­ experience­ that­ persists…a­ big­ problem­ if­ he­ gets­ that­ first­ experience­ wrong.­Allow­ pupils­ to­ ­experiment­and­fail­as­one­of­the­steps­on­the­path­to­success,­but­this­has­to­be­a­closely­ controlled strategy. If­ recall­of­facts­(such­as­times­table­facts)­and­procedures­(such­as­subtracting­from­ zero)­do­not­become­automatic,­then­there­is­less­mental­‘space’­left­to­do­the­main­task.­ This­compounds­the­effect­of­difficulties.­Select­‘easy’­numbers­when­introducing­new­ arithmetical procedures.

­Consider­giving­the­pupil­a­times­table­square­to­stick­into­the­back­of­his­exercise­book­ (so­that­he­has­to­make­some­effort­to­turn­to­the­information)­and­make­sure­he­can­ track successfully to the answers.

­Learning­is­usually­more­effective­if­it­is­presented­in­a­multisensory­way.­This­includes­ the­ use­ of­ concrete­manipulatives,­which­are­ often­phased­out­as­ being­‘too­young’­ for­secondary­pupils.­Manipulatives­may­be­used­as­demonstrations,­using­overhead­ ­projector­materials­or­via­PC­and­boards.­This­avoids­the­‘babyish’­image­when­used­by­

an­individual­learner.

­Money­ is­an­effective­manipulative­and­is­one­step­on­to­abstraction­from­a­directly­ ­proportional­manipulative­such­as­base­ten­blocks.­Also­it­is­likely­to­be­more­acceptable­ for older learners.

Concrete materials and manipulatives

As a teacher who started his career as a teacher of chemistry and physics one of the strange anomalies­of­teaching­mathematics­for­me­is­that,­unlike­science,­the­further­the­learner­ goes­up­the­mathematics­syllabus­the­less­likely­he­is­to­use­practical­materials.­Somewhere­ around about eight years old learners start to consider materials as patronising and babyish. Maybe this has something to do with the fact that some of these materials are bricks.

This­can­be­circumvented­by­the­teacher­using­the­materials­to­demonstrate­rather­ than­the­learner­engaging­in­discovery­‘play’­with­the­material­(those­teaching­philoso-phies­from­the­1960s­die­hard). I­think­that­one­of­the­most­important­decisions­a­teacher­makes­is­the­choice­of­ material(s)­to­illustrate­a­concept.­It­is­likely,­yet­again,­that­the­choice­will­be­depen-dent­on­the­learner.­At­least,­in­individual­work,­the­focus­should­be­the­learner­choosing­ a material that works for him rather than the teacher choosing his­favourite­material.

Materials­have­their­own­inherent­characteristics.­For­example,­a­metre­rule­will­give­ a­linear­image­of­units,­tens,­hundreds­and­thousands,­whereas­Dienes­(base­ten)­blocks­ will­give­one,­two­and­three­dimensional­images­of­units,­tens,­hundreds­and­thousands.­ Money­is­not­proportional­in­size­to­the­values­it­represents,­but­it­is­familiar­and­comes­ in­easy­number­values­based­on­1,­2­and­5. ­Inaccurate­intuitive,­answer-oriented­learners­are­at­risk. ­Dyslexic­pupils­do­not­adjust­quickly­to­changes­in­routine,­for­example­if­a­new­teacher­ expects­a­different­page­layout­in­exercise­books.

­ Sequential,­ formula-oriented­ learners­ with­ poor­ memories­ are­ at­ risk­ of­ failure­ in­ mathematics.

­Some ­pupils­are­intuitive,­answer-oriented­problem­solvers­who­may­not­learn­from­a­ step­by­step,­sequentially­oriented,­formula­dependent­teacher,­and,­of­course,­vice­versa.­ There­are­also­significant­implications­for­documentation­of­work.­Intuitive­workers­are­ usually­disinclined­to­document.­These­differences­in­cognitive­style­(thinking­through­ problems)­ are­ present­ in­ the­ whole­ school­ population,­ including­ teachers,­ but­ their­ affect­on­pupils­with­dyslexia,­dyspraxia­or­dyscalculia­(with­their­other­contributing­ problems)­is­likely­to­be­more­critical­(see­Chapter­4).

or­a­pupil­reads­the­problem­to­the­pup il­with­difficulties.­This­is­particularly­relevant­ for­coursework­and­investigations. ­ ­ Deal­with­the­pupils­who­are­forgetful­and­badly­organised.­Take­a­positive­attitude­and­ make­sure­they­have­the­information­and­equipment­they­need.­Parents­of­such­pupils­ have­usually­suffered­alongside­their­child­as­he­struggles­through­school.­You­could­ try­to­liaise­with­them,­for­example­by­giving­them­a­homework­timetable.­Remember,­ difficulties­may­be­familial. ­ Give ­homework­in­a­form­that­they­can­access.­For­example­check­the­vocabulary. ­ Make sure the homework is read to the pupil before he takes it home. Get high tech and provide­a­disc­so­that­work­can­be­done­on­computer­and­could­even­have­the­facility­of ­ the­PC­reading­the­questions­to­the­pupil. ­ Consider ­allowing­the­pupil­to­use­a­calculator­(with­all­the­cautions­I­know­you­have­ about­their­use)­or­a­number­square­or­a­table­square.

Marking

­

Mark­ new­ work­ before­ too­ many­ examples­ have­ been­ attempted.­ Do­ not­ let­ error­ patterns become ingrained.

­

Mark­diagnostically.­For­example,­the­pupil­may­have­used­the­correct­procedure,­but­ made­an­arithmetical­error.­Do­not­just­mark­work­‘wrong’.­Say­how­it­was­wrong­and­ what can be done to put it right.

Homework

If­a­dyslexic­ pupil ­is­having­difficulty­setting­out­work­on­the­page­discuss­giving­ him­ an­exercise­book­which­has­bigger/­smaller/­squares. The­reading­level­may­be­beyond­the­pupil.­If­a­book­cannot­be­replaced­in ­these­ economically­hard­times­either­provide­a­photocopied­‘translation’­or­make­sure­you­ ­Work­ sheets­and­text­book­layouts­can­be­overwhelming.­They­may­use,­for­exam-ple,­lots­of­small­print,­closely­spaced­or­fussy,­confused­pages­with­cartoons­and­ disjointed­text.­Try­providing­a­cover­sheet/­window­which­reduces­the­quantity­of­ material facing the pupil.

­

Books and worksheets

It­seems­to­be­stating­the­obvious,­but­the­material­must­make­the­concept­accessible.­ It­must­give­the­learner­an­unequivocal­image­of­the­idea­that­the­symbols­represent.­ The­material­and­the­symbols­must­be­related­in­the­learner’s­mind­(rather­than­just­the­ ­teacher’s­mind!)

The Catch 22 of Catch Up

If­a­pupil­falls­behind­he­or­she­will­have­been­working­more­slowly­than­their­peers.­To­ catch­up­he­or­she­will­have­to­progress­faster­than­their­peers.­It­is­possible.

A few golden rules

♦­ ­Don’t­create­anxiety.4 ♦­­ Experiencing­success­reduces­anxiety. ♦­­ Experiencing­failure­increases­anxiety. ♦­­ Understand­your­pupils­as­individuals. ♦­­ Teach­to­the­individual­in­the­group…also­known­as­the­‘Teach­more­than­one­way­to­ do­things’­rule. ♦­­ Remember­where­each­topic­leads­mathematically. ♦­­ Understanding­is­a­more­robust­outcome­than­just­recall. ♦­­ Try­to­understand­errors…don’t­just­settle­for­‘wrong’. ♦­­ Prevention­is­better­than­cure. ♦­­ All­the­above­rules­have­exceptions. ­ ­ ­ by­any­of­these­suggestions­and­many­will­be­advantaged.­The­suggestions­may­reduce­ some­of­the­learning­(special)­needs­in­your­classroom­and­even­prevent­the­onset­of­some­ problems. This­book­acknowledges­that­pupils­are­individuals.­I­have­long­had­a­suspicion­of­any­ scheme,­intervention­or­cure­that­claims­it­is­‘for­all’.­I­suspect­that­the­only­part­of­this­ book­that­is­‘for­all’­is­the­emphasis­on­understanding­each­pupil.­

Remember

Pupils­are­individuals.­Some­will­need­some­of­these­suggestions,­some­will­survive­without­ any­of­these­suggestions.­I­do­not­think,­however,­that­any­learner­will­be­disadvantaged­ ­Remember­ the­pupil­who­may­work­more­slowly­than­his­peers.­Consider­selecting­ fewer­examples­but­still­giving­the­breadth­of­experience. Be encouraging._­ Avoid­red­pens­and­big­crosses­and­scribbles.­(Try­green­and­small­and­neat­and,­better­ still,­constructive­comments­such­as­‘Small­addition­error­here,­rest­OK.’)

Gerstmann,­J.­(1957)­‘Some­notes­on­the­Gerstmann­syndrome’,­Neurology­7,­pp.­866–9.

Joffe,­ L.­ (1980)­ ‘Dyslexia­ and­ attainment­ in­ school­ mathematics:­ Part­ 1’,­ Dyslexia Review,­ 3­ 1,­ pp.­10–14.

Kosc,­L.­(1974)­‘Developmental­dyscalculia’,­Journal of Learning Disabilities,­7,­3,­pp.­46–59. Kosc,­L.­(1986)­Dyscalculia.­Focus on Learning Problems in Mathematics,­8,­3,­4.

Krutetskii,­V.A.­(1976)­in­Kilpatric,­J.­and­Wirszup,­I.­(eds)­The Psychology of Mathematical Abilities

in Schoolchildren.­Chicago,­University­of­Chicago­Press.

Magne,­O.­(1996)­Bibliography of Literature Dysmathematics Didakometry.­Malmo,­Sweden. Seligman,­M.­(1998)­Learned Optimism.­New­York,­Pocket­Books.

Sharma,­M.C.­(1986)­‘Dyscalculia­and­other­learning­problems­in­arithmetic:­a­historical­perspective’,­

Focus on Learning Problems in Mathematics­8,­3,­4,­pp.­7–45.

Sharma,­M.­(1990)­‘Dyslexia,­dyscalculia­and­some­remedial­perspectives­for­mathematics­learning­ problems.’­Math Notebook,­8,­7–10.

Stanovich,­K.E.­(1991)­‘The­theoretical­and­practical­consequences­of­discrepancy­definitions­of­ dyslexia.’­In­Snowling,­M.­and­Thomsom­M.­(eds),­Dyslexia: Integrating Theory and Practise. London,­Whurr.

Chinn,­S.J.­and­Ashcroft,­J.R.­(1992)­‘The­Use­of­Patterns’­in­T.R.Miles,­and­E.Miles,­(eds)­Dyslexia

and Mathematics,­London,­Routledge.

Chinn,­S.J.­and­Ashcroft,­J.R.­(1998)­Mathematics for Dyslexics: A Teaching Handbook.­London,­ Whurr.

Chinn,­S.J.,­McDonagh,­D.,­Van­Elswijk,­R,­Harmsen,­H.,­Kay,­J.,­McPhillips,­T.,­Power,­A.­and­ Skidmore,­ L.­ (2001)­ ‘Classroom­ studies­ into­ cognitive­ style­ in­ mathematics­ for­ pupils­ with­ ­dyslexia­in­special­education­in­the­Netherlands,­Ireland­and­the­UK’,­British Journal of Special

Education,­28,­2,­pp.­80–5.

Cohn,­ R.­ (1968)­ ‘Developmental­ dyscalculia’,­ Pediatric­ Clinics­ of­ North­ America,­ 15,­ (3),­ pp.­651–68.

DfES­ (2001)­ The National Numeracy Strategy: Guidance to Support Pupils with Dyslexia and

Dyscalculia.­London,­DfES.

References

Ashcraft,­M.,­Kirk,­E.P.­and­Hopkins,­D.­(1998)­‘On­the­cognitive­consequences­of­mathematics anxiety’,­in­Donlan,­C.­(ed.)­The Development of Mathematical Skills.­Hove,­The­Psychological Corporation.

Bakwin,­H.­and­Bakwin,­R.M.­(1960)­Clinical Management of Behaviour Disorders in Children. Philadelphia,­Saunders.

Butterworth,­B.­(2003)­The Dyscalculia Screener.­London,­NFER-Nelson.

Chinn,­ S.J.­ (1995)­ ‘A­ pilot­ study­ to­ compare­ aspects­ of­ arithmetic­ skill’,­ Dyslexia Review­ 4, pp. 4–7.

Chapter­2­

Factors that affect learning

This­chapter­is­about­some­of­the­factors­which­may­contribute­to­a­learner­having­difficulties­ with­ mathematics.­ Factors­ such­ as­ a­ poor­ short­ term­ memory­ can­ have­ a­ considerable­ impact on many of the topics that make up mathematics.

At­the­end­of­the­section­on­each­factor­I­have­left­space­for­the­reader­to­add­his­or­ her­own­suggestions.­I­could­never­claim­to­know­all­the­solutions,­especially­as­it­is­so­ important to remember that what works for one student may well not work for another. Hence­the­need­for­a­range­of­solutions,­to­be­used­in­response­to­the­individual­needs­of­ each learner.

Quite­often­the­‘Suggestions’­section­includes­the­advice­to­‘take­time­and­evaluate’.­ The­ culture­ of­ doing­ maths­ quickly­ can­ be­ totally­ opposite­ to­ the­ correct­ approach­ of­ appraising before reacting.

Some­suggestions­are­immediate­whilst­others­are­long­term.

Schools­and­colleges­should­work­towards­building­up­a­bank/­library­of­appropriate­ resources,­so­that­many­of­these­problems­can­be­addressed­from­materials­that­have­been­ prepared before.

Short term memory

The problem Suggestions

Some­children­will­have­weaker­short­term­memories­ than­their­peers.­This­may­be­a­developmental­lag­or­a­ persistent­problem.­What­is­relevant­here,­however,­is­the­ realisation that there will be a range of short term memory capacities among the pupils in any classroom.

This­will­impact­on­the­pupil’s­ability­just­to­keep­up­with­

the lesson in general. Do­not­give­lengthy­strings­of­instructions.­Imagine­someone­ ­giving­you­a­ten­digit­phone­number­ to­remember­by­presenting­it,­ once,­spoken­quickly­and­with­no­ breakdown into chunks of numbers. So­present­712563449­as­712­563­ 449­or­71­25­63­44­9­or­whatever­ chunking size pupils prefer. In­mental­arithmetic­it­may­be­a­challenge­for­them­to­